Mathematics 7.5 (aa) If θ is the angle between the line sinθ aa + bb + cc a + b + c A + B + C x x y y z z and the plane Ax + By + Cz + D 0, then a b c ax + by + cz + d (ab) Length of perpendicular from (x, y, z ) to the plane ax + by + cz + d 0 is a + b + c (ac) The equation of a sphere with center at the origin and radius a is r a or x + y + z a (ad) Equation of a sphere with center ( αβγ,, ) and radius a is (x α ) + (y β ) + (z γ ) a (ae) Vector equation of the sphere with center c and radius a is r c a or (r c) (r c) a (af) General equation of sphere is u + v + w d (ag) Equation of a sphere concentric with x + y + z + ux + vy + wz + d 0 whose center is ( u, v, w) and radius is x + y + z + ux + vy + wz + d 0 is x + y + z + ux + vy + wz +λ 0, where λ is a real number. Solved Examples JEE Main/Boards Example : Find the coordinates of the point which divides the join of P(,, 4) and Q(4,, ) in the ratio : (i) internally (ii) externally Sol: By using section formula we can obtain the result. Let R(x, y, z) be the required point 4 + 4 + ( ) (i) x ; y + 5 + 5 + 4 6 z + 5 4 6 So, the required point is R,, 5 5 5 4 ( ) (ii) x ; y 9 4 z 8 Therefore, the required point is R(. 9, 8) Example : Find the points on X-axis which are at a distance of 6 units from the point (,, ) Sol: Consider required point is P(x, 0, 0), therefore by using distance formula we can obtain the result. Let P(x, 0, 0) be a point on X-axis such that distance of P from the point (,, ) is 6 ( x) + ( 0) + ( 0) 6 (x ) + 4 + 9 4 (x ) x ± x ± Example : If a line makes angles αβγ,, with OX, OY, OZ, respectively, prove that sin α+ sin β+ sin γ Sol: Same as illustration. Let,m,n be the d.c. s of the given line, then cos α,m cos β,n cos γ cos α+ cos β+ cos γ ( sin α ) + ( sin β ) + ( sin γ ) sin α+ sin β+ sin γ Example 4: Projections of a line segment on the axes are, 4 and respectively. Find the length and direction cosines of the line segment. Sol: Let,m,n be the direction cosines and r be the length of the given segment, then r,mr,nr are the projections of the segment on the axes.
7.6 D Geometry Let,m,n be the direction cosines and r be the length of the given segment, then r,mr,nr are the projections of the segment on the axes; therefore r, mr 4, nr Squaring and adding, we get r ( + m + n ) + 4 + r 69 r length of segment And direction cosines of segment are 4 4, m and n r r r Example 5: Find the length of the perpendicular from the point (,, ) to the line through (6, 7, 7) and having direction ratios (,, ). Sol: By using distance formula i.e. (x x ) + m(y y ) + n(z z ), we can obtain required length. Direction cosines of the line are,, + + ( ) + + ( ) + + ( ) i.e.,, 7 7 7 AN Projection of AP on AB ( ) (6 ) + (7 ) + (7 ) 7 7 7 5 + 0 8 7 7 7 7 P(,, ) Also, B A(6, 7, 7) N d.r.s(,, -) AP (6 ) + (7 ) + (7 ) 5 + 5 + 6 66 PN AP AN 66 7 49 7 unit Example 6: Find the equation of the plane through the points A(,, ), B(, 4, ) and C(7, 0, 6) Sol: As we know, equation of a plane passing through the point (x, y, z ) is given by a(x x ) + b(y y ) + c (z z ) 0. The general equation of a plane through (,, ) is a(x ) + b(y ) + c(z + ) 0. (i) It will pass through B(, 4, ) and C(7, 0, 6) if a( ) + b(4 ) + c( + ) 0 or a + b + c 0. (ii) & a(7 ) + b(0 ) + c(6 + ) 0 or 5a b + 7c 0. (iii) Solving (ii) and (iii) by cross multiplication, we get a b c or a b c λ (say) 4 + 6 5 7 0 5 a 5 λ,b λ,c λ Substituting the values of a, b and c in (i), we get 5 λ(x ) + λ(y ) λ (z + ) 0 or,5(x ) + (y ) (z + ) 0 5x + y z 7, Which is the required equation of the plane. Example 7: Find the angle between the planes x + y + z 9 and x y + z 5 aa + bb + cc Sol: By using formula cosθ a + b + c a + b + c we can obtain the result. The angle between x + y + z 9 and x y + z 5 is given by cosθ aa + bb + cc + + + + a b c a b c ()() + ( ) + ()() π cosθ θ + + + ( ) + Example 8: Find the distance between the parallel planes x y + z + 0 and 4x y + 4z + 5 0 Sol: By making the coefficient of x, y and z as unity we will be get required result. Let P(x, y, z ) be any point on x y + z + 0, then, x y + z + 0 The length of the perpendicular from P(x,y,z )to4x y + 4z + 5 0 is 4x y + 4z + 5 (x y + z ) + 5 4 + ( ) + 4 6 ( ) + 5 6 6 (i)
Mathematics 7.7 Example 9: The equation of a line are 6x y + z. Find its direction ratios and its equation in symmetric form. Sol: The given line is 6x y + z 6 x y + (z ) x y+ z [We make the coefficients of x, y and z as unity] This equation is in symmetric form. Thus the direction ratios of the line are, and and this line passes through the point,,. Example 0: Find the image of the point (,, ) in the plane x y + 4z. Sol: Consider Q be the image of the point P(,, ) in the plane x y + 4z. Then PQ is normal to the plane hence direction ratios of PQ are,, 4. Let Q be the image of the point P(,, ) in the plane x y + 4z. Then PQ is normal to the plane. Therefore direction ratios of PQ are,, 4. Since PQ passes through P(,, ) and has direction ratios,, 4. Therefore equation of PQ is x y+ z r (say) 4 Let the coordination of Q be (r +, r,. 4r + ). Let R be the mid-point of PQ. Then R lies on the plane x y + 4z. The coordinates of R are r+ + r 4r+ +,, r + 6 r 4 or,,r + r + 6 r 4 + 4(r + ) r r So, the coordinates of Q are (0,, ) JEE Advanced/Boards Example : Find the equations of the bisector planes of the angles between the planes x y + z + 0 and x y + 6z + 8 0 and specify the plane bisecting the acute angle and the plane bisecting obtuse angle. Sol: As we know, Equation of the planes bisecting the angle between two given planes a x + b y + c z + d 0 and a x + b y + c z + d 0 are a x+ b y+ c z+ d a x+ b y+ c z+ d ± a + b + c a + b + c The two given planes are x y + z + 0 and x y + 6z + 8 0 (i) (ii) The equations of the planes bisecting the angles between (i) and (ii) are x y + z + x y + 6z + 8 ± + ( ) + + ( ) + 6 4x 7y + 4z + ± (9x 6y + 8z + 4) Hence the two bisector planes are 5x y 4z 0 (iii) and x y + z + 45 0 (iv) Now we find angle θ between (i) & (iii) We have, 5() + ( )( ) + ( 4) cosθ + ( ) + 5 + ( ) + ( 4) 4 π Thus the angle between (i) & (iii) is more than. Therefore, (iii) is the bisector of obtuse angle between 4 (i) and (ii) and hence (iv) bisects acute angle between them. Example : Find the distance of the point (,, ) from the plane x y + z 5 measured parallel to the line whose direction cosines are proportional to,, 6. Sol: By using distance formula we can obtain required length, Equation of line through (,, ) parallel to the line with d.r. s,, 6 is x y+ z r 6 Any point on it is ( + r, + r, 6r) Line (i) meets the plane x y + z 5. If + r ( + r) + ( 6r) 5 ; i.e. if r 7 (i)
7.8 D Geometry 9 5 Point of intersection is,, 7 7 7 whose distance from (,, ) is 9 5 + + + 7 7 7 4 9 6 + + 49 49 49 49 49 Example : Show that the lines x y+ z (i) 5 and x+ y z+ (ii) 4 do not intersect. Also find the shortest distance between them. Sol: If x x y y z z m n 0 m n then the lines do not intersect each other. And using distance formula we will be get required shortest distance. Points on (i) and (ii) are (,, ) and (,, ) 5 respectively and their d.c. s are,, 8 8 8 4 and,, respectively. 9 9 9 x x y y z z m n m n 5 8 9 4 + 5 8 8 8 4 9 9 9, [ ( 4 5) + (0 + 6) (9 8)] 0 8 9 Hence the given lines do not intersect. Any point P on (i) is ( + r, r,, 5r + ) and a point on (ii) is Q(4r, r +, r ) Direction ratios of PQ are (4r r, r r +, r 5r ) If PQ is perpendicular to (i) and (ii), we have (4r r ) + (r r + ) + 5( r 5r ) 0 & 4(4r r ) + (r r + ) ( r 5r ) 0 i.e. 8r 8r 5 0 & 9r 8r 0 r r Solving them, 76 0 45 + 6 08 r 44 49,r 08 08 57 88 095 Points P and Q are +,, + 08 08 08 76 88 and, +, 08 08 08 We can find the distance PQ by distance formula which is the shortest distance. Example 4: Find the angle between the lines whose direction ratios satisfy the equations : + m + 5n 0, 6mn n+ 5m 0 Sol: Here, cosθ aa' + bb' + cc' a + b + c a' + b' + c' The given equations are + m + 5n 0 (i) and 6mn n+ 5 m 0 (ii) From (i), we have m 5n (iii) Putting in (ii), we get 6( 5n) n n+ 5 ( 5n) 0 0n + 45n + 5 0 n + n + 0 (n + )(n + ) 0 Either n or n If If n, then from (iii), m n n, then from (iii), m n Thus the direction ratios of two lines are n, n, n and n, n, n i.e.,, and,, If θ is the angle between the lines, then.( ) +. + ( ). + cosθ. + 4+. 4+ + 6. 6 6 Example 5: Find the equation of the plane through the intersection planes x + y + 4z 5, x y + z and parallel to the straight line having direction cosines (,, )..
Mathematics 7.9 Sol: By using formula of family of plane, we will get the result. Equation of plane through the given planes is x + y + 4z 5 +λ(x y + z ) 0 i.e. ( + λ )x + ( λ )y + (4 + λ )z + ( 5 λ ) 0 This plane is parallel to the given straight line. ( + λ ) + ( λ ) + ( )(4 + λ ) 0 λ+ λ 4 λ 0 6λ λ Equation of required plane is 7 7 x + y + z 0 x + 7y + 6z 7 Example 6: Prove that the lines x+ y + z+ 5 x y 4 z 6 and 5 7 4 7 are coplanar. Also, find the plane containing these two lines. Sol: As similar to example. x x y y z z We know the lines m n x x y y z z and are coplanar if m n x x y y z z m n 0 m n and the equation of the plane containing these two lines is x x y y z z m n 0 m n Here x,y,z 5,x,y 4,z 6,,m 5,n 7,,m 4,n 7 x x y y z z m n m n 98 + 77 0 7 5 7 4 7 So, the given lines are coplanar, The equation of the plane containing the lines is x+ y+ z+ 5 5 7 0 4 7 or(x + )(5 8) (y + )( 7) + (z + 5)( 5) 0 o r x y + z 0 Example 7: Find the equation of the plane passing through the lines of intersection of the planes x y 0 and z y 0 and perpendicular to the plane 4x + 5y z 8. Sol: Here by using the family of plane and formula of two perpendicular plane we will get the result. The plane x y + k(z y) 0 x ( + k) y + kz 0 is perpendicular to the plane 4x + 5y z 8.4 ( + k).5 + k( ) 0 4k k 4 Thus the required equation is x y + (z y) 0 4 Example 8: Show that the lines 8x 7y + 9z 0 x+ 5 y+ 4 z 7,x + y + z 0 x y + z are coplanar and find the equation to the plane in which they lie. Sol: By using the condition of coplanarity of line, we will get given lines are coplanar or not. And after that by using general equation of the plane we can obtain required equation of plane. The general equation of the plane through the second line is x + y + z + k(x y + z ) 0 x(+ k) + y( k) + z( + k) k 0 ; K being the parameter This contains the first line only if 9 ( + k) + ( k) ( + k) 0 k 4 Hence the equation of the plane which contains the two lines is x 9y + x 5 0 This plane clearly passes through the point ( 5, 4,7)
7.0 D Geometry JEE Main/Boards Exercise Q. Direction cosines of a line are,, 6, direction ratios. 7 7 7 find its Q.5 Find the intercepts cut by the plane x y + 4z 0 on axes. Q.6 Direction ratios of a line are,,. Find its direction cosines. Q. Find the direction ratios of a line passing through the points (,, 0) and (,, ). Q. Find the angle between the lines x y+ 5 z and. 0 Q.4 Find the equation of a line parallel to the vector i ˆ ˆj kˆ and passing through the point (,,). Q.5 Write the vector equation of a line whose Cartesian x+ y z equation is +. 4 5 Q.6 Write the Cartesian equation of a line whose vector equations is r (i ˆ+ ˆj 5k) ˆ +λ( i ˆ+ ˆj + k). ˆ Q.7 Find the value of p, such that the line and p 5 are perpendicular to each other. Q.8 Write the Cartesian equation of the plane r(i ˆ+ ˆj + 5k) ˆ 7. Q.9 Write the vector equation of plane x y 4z + 7 0. Q.0 Find the vector, normal to the plane r.(i ˆ 7k) ˆ + 5 0. Q. Find the direction ratios of a line, normal to the plane 7x + y z. Q. Find the angle between the line and the plane x z + 4 0. x+ y z 4 5 7 Q. Find the distance of the plane x + y + z + 7 0 from origin. Q.4 Find the distance of the plane x y + z 0 from the point (,, ). Q.7 Find the direction cosines of y-axis. Q.8 Find the direction ratio of the line x+ y z. 5 Q.9 Find the angle between the planes r.(i ˆ ˆj k) ˆ andr.(i ˆ 6ˆj + k) ˆ 0. Q.0 Find the angle between the line r (i ˆ ˆj + k) ˆ + +λ(i ˆ 6j ˆ+ k) ˆ and the plane r.(i ˆ+ ˆj + k) ˆ + Q. Find the direction cosines of the two lines which are connected by the relations 5m + n 0and7 + 5m n 0. Q. Prove that, the line passing through the point (,, ) and (,, ) is perpendicular to the line passing through the points (,, 5) and (,, ). Q. Find the coordinates of the foot of the perpendicular drawn from the point (,, ) to the line joining the points (, 4, 6) and (5, 4, 4). Q.4 If a variable line in two adjacent positions has direction cosines,m,n and +δ,m+δ m,n+δn, prove that the small angle δθ between two position is given by ( δθ ) ( δ ) + ( δ m) + ( δn). + + m + m + m n + n + n Q.5 Verify that,, can be taken as direction cosines of a line equally inclined to three mutually perpendicular lines with direction cosines,m,n ;,m,n and,m,n Q.6 Find the equations of line through the point (, 0, ) and parallel to the planes x + y 0 and y z 0.
Mathematics 7. Q.7 Find the equations of the planes through the intersection of the planes x + y + 6 0 and x y 4z 0 whose perpendicular distance from the origin is equal to. Q.8 Find the equation of the plane through the points (,, ) and (,, ) and perpendicular to the plane x + y + z 5. Q.9 Find the distance of the point (, 5, 0) from the plane x y + z 5 measured parallel to the line x y+ z. 4 Q.0 Find the vector and Cartesian forms of the equation of the plane passing through (,, 4) and parallel to the line r ˆi + j ˆ 4k ˆ +λ (i ˆ+ j ˆ+ 6k) ˆ and r ˆi j ˆ+ 5k ˆ +µ (i ˆ+ ˆj k). ˆ Q. If straight line having direction cosines given by a + bm + cn 0 and fmn + gn+ h m 0 are g perpendicular, then prove that f + + h 0. a b c Q. Prove that, the lines x ay + b, z cy + d and x a y + b, z c y + d are perpendicular to each other, if aa + cc + 0. Q. Find the equation of the plane passing through the intersection of the planes 4x y + z 0 and x + y z 4 and parallel to the line with direction ratios,,. Find also the perpendicular distance of (,, ) from this plane. Q.4 The foot of the perpendicular drawn from the origin to the plane is (, 5, 7). Find the equation of plane. Q.5 Find the equation of a plane through (,, ) and perpendicular to the planes x + y z and 5x 4y + z 5. Q.6 Find the angle between the lines whose direction cosines are given by equations + m + n 0; + m n 0 Q.7 Find the equation of the line which passes through (5, 7, ) and is parallel to the line of intersection of the planes x y 5 0 and 9y z + 6 0. Q.8 Prove that, the plane through the points (,, ), (,, ) and ( 7,, 5) is perpendicular to xz-plane. Q.9 Find the length and coordinates of the foot of perpendicular from points (,, ) to the plane x y + 4z + 5 0. Q.40 Find the vector equation in the scalar product form, of the plane passing through the points (, 0, ), (,, ) and parallel to line r ˆi + ˆj +λ(i ˆ ˆj + k). ˆ Q.4 Find the distance between the parallel planes x y + z 4 0 and 6x y + 9z + 0. Q.4 Prove that, the equation of a plane. Which meets the axes in A, B, and C and the given centroid of triangle ABC is the point ( αβγ,, ), is + +. α β γ Q.4 Find the equation of the plane passing through the origin and the line of intersection of the planes x y + z + 4 0 and x y + z + 0. Q.44 Prove that, the line x + y z 6 0, x + y z 8 0 is parallel to the plane y 0. Find the coordinates of the point where this line meets the plane x 0. Q.45 Find the equation of the plane through the line ax + by + cz + d 0, a x + b y + c z + d 0 and parallel y to the line x z. m n Q.46 Find the equation of a plane parallel to x-axis and has intercepts 5 and 7 on y and z-axis, respectively. Q.47 A variable plane at a constant distance p from origin meets the coordinate axes in points A, B and C, respectively. Through these points, planes are drawn parallel to the coordinate planes, prove that locus of point of intersection is + +. p Q.48 Find the value of λ, for which the points with position vectors ˆ i ˆ j + k ˆ and i ˆ +λ ˆ j + k ˆ are equidistant from the plane r.(5i ˆ+ ˆj 7k) ˆ + 9 0. Q.49 Find the equation of a plane which is at a distance of 7 units from the origin and which is normal to the vector i ˆ+ 5j ˆ 6kˆ Q.50 Find the vector equation of the plane, r ˆi ˆj +λ (i ˆ+ ˆj + k) ˆ +µ (4i ˆ j ˆ+ k) ˆ in the scalar product from.
7. D Geometry Exercise Single Correct Choice Type Q. The sum of the squares of direction cosines of a straight line is (A) Zero (C) (B) Two (D) None of these Q. Which one of the following is best condition for the plane ax + by + cz + d 0 to intersect the x and y axes at equal angle (A) a b (B) a b (C) a b (D) a + b Q. The equation of a straight line parallel to the x-axis is given by x a y b z c x a y b z c (A) (B) 0 x a y b z c x a y b z c (C) (D) 0 0 0 0 Q.4 A straight line is inclined to the axes of x and z at angles 45 and 60 respectively, then the inclination of the line to the y-axis is (A) 0 (B) 45 (C) 60 (D) 90 Q.5 The coordinates of the point of intersection x+ y z of the line + + with the plane x + 4y + 5z 5 (A) (5, 5, 4) (B) (, 4, 5) (C) (,, ) (D) (,, 0) Q.6 Perpendicular is drawn from the point (0,, 4) to the plane x y + z 0. The coordinates of the foot of the perpendicular are (A) 8 6,, (B) 8,, 6 (C) 8,, 6 (D) 8,, 6 Q.7 The equation of the plane through the line of intersection of the planes x + y z 4 0 and x + 5z 4 0 which cuts off equal intercepts from the x-axis and y-axis is (A) x + y 8z + 8 0 (B) x + y 8z 8 0 (C) x y 8z 8 0 (D) x + y 8z 8 0 Q.8 The symmetric form of the equation of the line x + y z, x y + z is x+ y z (A) + (B) 5 x y z x y z (C) (D) 5 5 x y z 4 Q.9 The line is parallel to the plane (A) x + y + z + 0 (B) x y z (C) x y + z 0 (D) x + y z 0 Q.0 The vertices of the triangle PQR are (,, ), (,, ) and ( 4, 0, ). The area of the triangle is (A) 8 (B) 8 (C) 4 (D) Q. Equation of straight line which passes through the point P(, 0, ) and Q(,, 4) is x y+ z 4 x y z+ (A) (B) x / y z 4 (C) + x y z+ (D) Q. A point moves so that the sum of the squares of its distances from the six faces of a cube given by x ±, y ±, z ± is 0 units. The locus of the point is (A) x + y + z (B) x + y + z (C) x + y + z (D) x + y + z Q. The points (0,, ), ( 4, 4, 4), (4, 5, ) and (, 9, 4) are (A) Collinear (C) Forming a square (B) Coplanar (D) None of these Q.4 The equation of the plane containing the line x α y β z γ is a(x α ) + b(y β ) + c(z γ ) 0, m n where a + bm + cn is equal to (A) (B) (C) (D) 0 Q.5 The reflection of the plane x + y + 4 z 0 in the plane x y + z 0 is the plane (a) 4x y + z 5 0 (b) x y + z 5 0 (c) 4x + y z + 5 0 (d) None of these
Mathematics 7. Previous Years Questions Q. The value of k such that in the plane x 4 y z k lies x 4y + z 7, is (00) (A) 7 (B) 7 (C) No real value (D) 4 Q. If the lines r a +µ b and x y k z intersect, then the value of k is (004) (A) a (B) 9 (C) (D) 9 Q. A variable plane + + at a unit distance a b c from origin cuts the coordinate axes at A, B and C. Centroid (x, y, z) satisfies the equation + + K. The value of K is (005) (A) 9 (B) (C) /9 (D) / Fill in the Blanks for Q.4 and Q.5 Q.4 The area of the triangle whose vertices are A(,, ), B(,, ), C(,, ) is (98) Q.5 The unit vector perpendicular to the plane determined by P(,, ), Q(, 0, ) and R(0,, ) is.. (98) Q.6 A plane is parallel to two lines whose direction ratios are (, 0, ) and (,, 0) and it contains the point (,, ). If it cuts coordinate axes at A, B, C. Then find the volume of the tetrahedron OABC. (004) Q.7 Find the equation of the plane containing the line x y + z 0, x + y + z 5 and at a distance of 6 from the point (,, ). (005) x y z 4 Q.8 If the line + + lies in the plane, x + my z 9, then + m is equal to: (06) (A) 8 (B) 5 (C) (D) 6 Q.9 The distance of the point (, 5,9) from the plane x y+ z 5measured along the line x y zis (06) (A) 0 (B) 0 (C) 0 (D) 0 Q.0 The distance of the point (, 0, ) from the point x y+ z of intersection of the line and the 4 plane x y + z 6, is: (05) (A) 8 (B) (C) (D) 4 Q. The equation of the plane containing the line x 5y + z + ; x+ y+ 4 z 5 and parallel to the plane x + y + 6z is (05) (A) x + y + 6z 7 (B) x + y + 6z 7 (C) x + 6y + z (D) x + 6y + z Q. The number of common tangents to the circles (05) (A) Meats the curve again in the second in the second quadrant (B) Meats the curve again in the third quadrant (C) Meets the curve again in the fourth quadrant (D) Does not meet the curve again x y z 4 Q. The image of the line in the 5 plane x y + z + is the (04) x+ y 5 z (A) x+ y (B) 5 z 5 5 x+ y 5 z (C) x y 5 z (D) 5 5 Q.4 The angle between the lines whose direction cosines satisfy the equations l+m+n0 and l m +n is (04) π π π π (A) (B) (C) (D) 4 6 Q.5 If the lines x y z 4 x y 4 z 5 and k k are coplanar, then k have (0) (A) Exactly one value (B) Exactly two value (C) Exactly three values (D) Any value Q.6 An equation of a plane parallel to the plane x y + z 5 0 and at a unit distance from the origin is (0) (A) x y + z 0 (B) x y + z + 0 (C) x y + z 0 (D) x y + z + 5 0
7.4 D Geometry y z Q.7 If the angle between the line x and λ 5 the plane x + y + z 4 is cos, then λ equals 4 (0) (A) (B) 5 (C) 5 (D) (A) Statement-I is true, statement-ii is true; statement-ii is not a correct explanation for statement-i (B) Statement-I is true, statement-ii is false. (C) Statement-I is false, statement-ii is true (D) Statement-I is true, statement-ii is true, statement-ii is a correct explanation for statement-i Q.8 Statement I: The point A(,0,7) image of the point B(,6,) in the line Statement-II: The line: is the mirror x y z x y z bisects the line segment joining A(,0,7) and B(,6, ) (0) JEE Advanced/Boards Exercise Q. Points X and Y are taken on the sides QR and RS respectively, of parallelogram PQRS, so that QX 4 XR and RY 4 YS. The line XY cuts the line PR at Z. prove that PZ PR. 5 Q. Given three points on the xy plane on O(0, 0), A(, 0) and B(, 0). Point P is moving on the plane satisfying the condition (PA PB) + (OA OB) 0. If the maximum and minimum values of PA PB are M and m, respectively then find the value of M + m. Instruction for questions to 6. Suppose the three vectors, a,b, c on a plane satisfy the condition that a b c a+ b ;c is perpendicular to a and b c > 0, then Q. Find the angle formed by a + b and b. Q.4 If the vector c is expressed as a linear combination λ a+µ b then find the ordered pair m m n n, and. sin( θ/ ) sin( θ/ ) sin( θ/ ) Q.5 For real number x, y the vector p xa+ yc satisfies the condition 0 p a and 0 p b. Find the maximum value of p. c Q.6 For the maximum value of x and y, find the linear combination of p in terms of a and b. Q.7 If O be the origin and the coordinates of P be (,, -), then find the equation of the plane passing through P and perpendicular to OP. Q.8 Given non zero number x,x,x ; y,y,y and z,zandz (i) Can the given numbers satisfy x x x xx + yy + zz 0 y y y 0 and xx + yy + zz 0 z z z xx + yy + zz 0 (ii) If x > 0 and y < 0 for all I,, and P (x,x,x ); Q(y,y,y ) and O(0, 0, 0) can the triangle POQ be a right angled triangle? Q.9 ABCD is a tetrahedron with pv s of its angular points as A( 5,,5);B(,,);C(4,,) and D(,, ). If the area of the triangle AEF where the quadrilaterals ABDE and ABCF are parallelogram is S, then find the value of S.
Mathematics 7.5 Q.0 If x, y are two non-zero and non-collinear vectors satisfying [(a ) α + (b ) α+ c]x + [(a ) β + (b ) β+ c]y + [(a ) γ + (b ) γ+ c](x y) 0 where αβγ,, are three distinct real numbers, then find the value of (a + b + c ). Q. Find the distance of the point (-, -5, -0) from the point of intersection of the line r i ˆ ˆj + kˆ +λ ( i ˆ+ 4ˆj + kˆ) and the plane r. ˆi ˆj + kˆ 5 ( ) Q. Find the equations of the straight line passing through the point (,, ) to intersect the straight line x + (y ) x + 4 and parallel to the plane x + 5y + 4z 0. Q. Find the equations of the two lines through the x y z origin which intersect the line at an π angle of. Exercise Single Correct Choice Type Q. If P(,, 6) and Q(, 4, 5) are two points, the direction cosines of line PQ are 7 7 (A),, (B),, 7 7 7 7 7 7 7 7 (C),, (D),, 7 7 7 7 7 7 Q. The ratio in which yz-plane divide the line joining the points A(,, 5) and B(, 4, 6) is (A) : (B) : (C) : (D) : Q. The value of λ for which the lines x + y + z + 5 0 x + y z and x y λ z 0 7x + 0y 8z are perpendicular to each other is (A) (B) (C) (D) Q.4 The ratio in which yz-plane divides the line joining (, 4, 5) and (, 5, 7) (A) : (B) : (C) : (D) : Q.5 A line makes angle αβγδ,,, with the four diagonals of a cube then cos α+ cos β + cos γ+ cos δ is equal to (A) (B) 4/ (C) ¾ (D) 4/5 Q.6 A variable plane passes through a fixed point (a, b, c) and meets the coordinate axes in A, B, C. The locus of the point common to plane through A, B, C parallel to coordinate planes is (A) ayz + bzx + cxy xyz (B) axy + byz + czx xyz (C) axy + byz + czx abc (D) bcx + acy + abz abc Q.7 The equation of the plane bisecting the acute angle between the planes x y + z + 0 and x y + 6z + 8 0 (A) x y + z + 45 0 (B) 5x y 4z (C) 5x y 4z + 45 0 (D) x y + z + 0 Q.8 The shortest distance between the two straight x 4/ y+ 6/5 z / lines and 4 5y + 6 z x 4 is 8 9 5 (A) 9 (B) (C) 0 (D) 6 0 Q.9 The equation of the straight line through the origin parallel to the line (b + c)x + (c + a)y + (a + b) z k (b c)x + (c a)y + (a b)z is (A) b c c a a b (B) b c a (C) a bc b ca c ab (D) None of these Assertion Reasoning Type Q.0 Consider the following statements Assertion: The plane y + z + 0 is parallel to x-axis. Reason: Normal to the plane is parallel to x-axis.
7.6 D Geometry (A) Both A and R are true and R is the correct (B) Both A and R are true and R is not a correct explanation of A (C) A is true but R is false (D) A is false but R is true Q.5 Consider three planes x y 8 z AB : λ x+ y+ 7 z 6 CD : µ 4 L (λ+, λ+ 8, λ+ ) and Previous Years Questions Q. A plane passes through (,, ) and is perpendicular to two planes x y + z 0 and x y + z 4, then the distance of the plane from the point (,, ) is (006) (A) 0 (B) (C) (D) Q. Let P(,, 6) be a point in space and Q be a point on the line r (i ˆ ˆj + k) ˆ +µ ( i ˆ+ ˆj + 5k) ˆ. Then the value of µ for which the vector PQ is parallel to the plane x 4y + z is (009) (A) 4 (B) (C) 4 8 (D) 8 Q. A line with positive direction cosines passes through the point P(,, ) and makes equal angles with the coordinate axes. The line meets the plane x + y + z 9 at point Q. The length of the line segment PQ equals (009) (A) (B) (C) (D) For the following question, choose the correct answer from the codes (A), (B), (C) and (D) defined as follows. (A) Statement-I is true, statement-ii is also true; statement-ii is the correct explanation of statement-i (B) Statement-I is true, statement-ii is also true; statement-ii is not the correct explanation of statement-i. (C) Statement-I is true; statement-ii is false. (D) Statement-I is false; statement-ii is true Q.4 Consider the planes x 6y z 5 and x + y z 5. Statement-I: The parametric equations of the line of intersection of the given planes are x + 4t, y + t, z 5t. Statement-II: The vectors 4i ˆ+ j ˆ+ 5kˆ is parallel to the line of intersection of the given planes. (007) Let L, L, L be the lines of intersection of the planes P and P, P and P, P and P, respectively. Statement-I : At least two of the lines L, L and L are non-parallel. Statement-II : The three planes do not have a common point (008) Paragraph for Q.6 to Q.8 Read the following passage and answer the questions. Consider the lines x+ y z x y z L : + +,L : + (008) Q.6 The unit vector perpendicular to both L and L is ˆi + 7 ˆj + 7kˆ (A) 99 ˆi + 7 ˆj + 5kˆ (C) 5 (B) (D) ˆi 7 ˆj + 5kˆ 5 7i ˆ 7j ˆ kˆ 99 Q.7 The shortest distance between L and L is (A) 0 (B) 7 (C) 4 5 (D) 7 5 Q.8 The distance of the point (,, ) from the plane passing through the point (-, -, -) and whose normal is perpendicular to both the lines L and L is (A) 75 (B) 7 75 (C) 75 (D) 75 Match the Columns Match the condition/expression in column I with statement in column II. Q.9 Consider the following linear equations ax + by + cz 0, bx + cy + az 0, cx + ay + bz 0 (007) Column I (A) a+ b+ c 0 and a + b + c ab + bc + ca Column II (p) The equations represent planes meeting only at a single point
Mathematics 7.7 (B) a+ b+ c 0 and a + b + c ab + bc + ca (C) a+ b+ c 0 and a + b + c ab + bc + ca (D) a+ b+ c 0 and a + b + c ab + bc + ca (q) The equation represent the line x y z (r) The equations represent identical planes (s) The equations represent the whole of the three dimensional space Q.4 In R, consider the planes P : y0 and P : x+z. Let P be a plane, different from P and P, which passes through the intersection of P and P. If the distance of the distance of the point (0,, 0) from P is and the distance a point (α, β, γ) from p is, then which of the following relations is (are) true? (05) (A) α+β+ γ+ 0 (B) α β+ γ+ 4 0 (C) α+β γ 0 0 (D) α β+ γ 8 0 Q.0 (i) Find the equation of the plane passing through the points (,, 0), (5, 0, ) and (4,, ). (ii) If P is the point (,, 6), then the point Q such that PQ is perpendicular to the plane in (a) and the mid point of PQ lies on it. (00) Q. T is a parallelepiped in which A, B, C and D are vertices of one face and the face just above it has corresponding vertices A, B, C, D, T is now compressed to S with face ABCD remaining same and A, B, C, D shifted to A, B, C, D in S. the volume of parallelepiped S is reduced to 90% of T. Prove that locus of A is a plane. (00) Q. Consider a pyramid OPQRS located in the first octant ( x 0,y 0,z 0) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP.The point S is directly above the mid-point T of diagonal OQ such that TS.Then (06) (A) The acute angle between OQ and OS is π (B) The equation of the plane containing the triangle OQS is x y 0 (C) The length of the perpendicular from p to the plane containing the triangle OQS is (D) The perpendicular distance from O to the straight line containing RS is 5,,7 with respect to the plane x y+ x.then equation of the plane passing through P and containing the straight line is (06) (A) x + y z 0 (B) x + z 0 (C) x 4y + 7z 0 (D) x y 0 Q. Let P be the image of the point ( ) Q.5 In R let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P : x+ y z+ 0 and P :x y+ z 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P. Which of the following points lie (s) on M? (05) 5 (A) 0,, 6 5,0, (C) 6 6 Q.6 From a point p (,, ) (B),, 6 6 (D),0, λλλ perpendiculars PQ and PR are drawn respectively on the lines y x,z and y x,z. If p is such that QPR is a right angle, then the possible value(s) of λ is(are) (04) (A) (B) (C) - (D) Q.7 Perpendiculars are drawn from points on the line x+ y+ z to the plane x + y + z. The feet of perpendiculars lie on the line (0) (A) (C) x y z (B) 5 8 x y z (D) 4 7 x y z 5 x y z 7 5 y z Q.8 Two lines L : x 5, α and y z L :x α, are coplanar. α The α can take value(s) (0) (A) (B) (C) (D) 4 x y z+ Q.9 Consider the lines L :, x 4 y+ z+ L : and the planes
7.8 D Geometry P : 7x + y + z,p : x + 5y 6z 4. Let ax + by + cz d be the equation of the plane passing through the point of intersection of lines L and L.and perpendicular to planes P and P Match List I with List II and select the correct answer using the code given below the list: (0) Codes: List I List II p. a. q. b. - r. c. s. d 4. - p q r s (A) 4 (B) 4 (C) 4 (D) 4 Q.0 The point P is the intersection of the straight line R,,4 with the plane joining the point Q (,,5 ) and ( ) 5x 4y z. If S is the foot of the perpendicular drawn from the point T(,,4) to QR, then the length of the line segment PS is (0) (A) (B) (C) (D) Q. The equation of a plane passing through the line of intersection of the planes x + y + z and x y+ z and at a distance from the point (,, ) is (0) (A) 5x y + z 7 (B) x + y (C) x+ y+ z (D) x y x t Q. If f ( x) e (t ) ( t ) dt for all x ( 0, ) 0 (A) f has a local maximum at x (B) f is decreasing on (, ) then (C) There exists some c ( 0, ) such that f (c)0 (D) f has local minimum at x (0) Q. If the distance between the plane Ax y + z d and the plane containing the lines x y z x y z 4 and is 6, 4 4 5 then d is (00) Q.4 If the distance of the point P(,,) from the plane x + y z α, where α> 0, is 5, then the foot of the perpendicular from P to the plane is (00) 8 4 7 (A),, 0 (C),, 4 4 (B),, 5 (D),, Q.5 Two adjacent sides of a parallelogram ABCD are given by AB i + 0 j + k and AD + i j + k The side AD is rotated by an acute angle a in the plane of the parallelogram so that AD becomes AD'. If AD makes a right angle with the side AB, then the cosine of the angle a is given by (00) (A) 8 9 (B) 7 9 (C) 9 (D) 4 5 9
Mathematics 7.9 MASTERJEE Essential Questions JEE Main/Boards Exercise Q.5 Q.0 Q. Q.9 Q.6 Q.40 Q.4 Q.47 Q.49 Q.50 Exercise Q. Q.8 Q. Q. Q.4 Previous Years Questions Q. Q.6 JEE Advanced/Boards Exercise Q. Q.5 Q.8 Q.0 Q. Exercise Q. Q.5 Q.6 Q.7 Q.9 Previous Years Questions Q. Q.5 Q.6 Q.9 Q. Answer Key JEE Main/Boards Exercise 7 Q. <,, 6> Q. <,, > Q. cos 70 Q.4 r ( ˆi + ˆj + k) ˆ +λ(i ˆ ˆj k) ˆ Q.5 r ( i ˆ+ ˆj k) ˆ +λ (i ˆ+ 4j ˆ+ 5k) ˆ Q.6 x y z+ 5 Q.7 Q.8 x + y +5z 7 Q.9 r(i ˆ ˆj 4k) ˆ + 7 0 Q.0 i ˆ 7j ˆ Q. <7,, > Q. sin 90 Q. 7 Q.4 Q.5 4, 6, Q.6 4 4 4 Q.7 <0,, 0> Q.8 <,, 0>
7.40 D Geometry Q.9 cos Q.0 Q. (, 4, 5) Q.6 sin 7 x y z Q..,,,, 4 4 4 6 6 6 Q.7 x y x 0; x + y z + 0 Q.8 x + y z + 0 Q.9 units Q.0 9x 8y + z + 0 Q. 5y 5z 6 0, 5 Q.4 x + 5y + 7z 78 Q.5 5x + 9y + z 8 0 π x 5 y+ 7 z+ 5 6 Q.6 4 Q.7 Q.9,,, 9 6 Q.40 r(4i ˆ ˆj k) ˆ 6 Q.4 5 4 4 Q.4 x + y 5z 0 a + bm + cn Q.44 (0,, ) Q.45 (ax + by + cz + d) (a'x + b'y + c'z + a') 0 (a' + b'm+ c'm) Q.46 7y + 5z 5 Q.48 λ, 6 Q.49 r(i ˆ+ 5ˆj 6k) ˆ 7 70 0 Q.50 r(5i ˆ+ ˆj 6k) ˆ 4 Exercise Single Correct Choice Type Q. C Q. A Q. D Q.4 C Q.5 A Q.6 B Q.7 B Q.8 D Q.9 C Q.0 A Q. D Q. B Q. B Q.4 D Q.5 A Previous Years Questions (i ˆ+ ˆj + k) ˆ Q. A Q. B Q. A Q.4 ssq. units Q.5 ± Q.6 9 cu unit 6 Q.7 x y + z 0 and 6x + 9y + 9z 05 0 Q.8 C Q.9 A Q.0 C Q. B Q. B Q. A Q.4 A Q.5 B Q.6 A Q.7 D Q.8 B JEE Advanced/Boards Exercise Q. 4 Q. π Q.4, Q.5 Q.6 p (a + b) Q.7 x + y z 4 Q.8 No, No Q.9 0 Q.0 Q. Q. x y z Q. or
Mathematics 7.4 Exercise Single Correct Choice Type Q. B Q. A Q. D Q.4 A Q.5 B Q.6 A Q.7 A Q.8 C Q.9 C Assertion Reasoning Type Q.0 C Previous Years Question Q. D Q. A Q. C Q.4 D Q.5 D Q.6 B Q.7 D Q.8 C Q.9 A r;b q;c p;d s Q.0 (a) x + y z (b) Q(6, 5, ) Q. B, C, D Q. C Q.4 B, D Q.5 A, B Q.6 C Q.7 D Q.8 A, D Q.9 A Q.0 A Q. A Q. B, C Q. 6 Q.4 A Q.5 B Solutions JEE Main/Boards Exercise 6 Sol : l m n 7 7 7 Direction ratios are <,, 6> Sol : [,, 0] & [,, ] Direction ratios, +, 0 <,, > Sol : and x 0 <,, 0> and <,, >. +. + 0. cosθ 5 4 Sol 4: y + 5 7 70 x+ y z t r ˆi + ˆj + k ˆ + t ( i ˆ ˆj kˆ) z 7 θ cos 70 Sol 5: r ˆi + ˆj k ˆ +λ (i ˆ + 4ˆj + 5k) ˆ Sol 6: xi ˆ + yj ˆ + zk ˆ ( λ )i ˆ + ( + λ )j ˆ + ( 5 + λ)kˆ x y z + 5 Sol 7: cosθ 0. +.5 + p. + 4p 0 p Sol 8: (xi + yj + zk). (i + j + 5k) 7 x+ y + 5z 7 Sol 9: x y 4z 7; r(i j 4k) 7 r(i ˆ ˆj 4k) ˆ + 7 0 Sol 0: x 7y 5 Direction ratios of normal to plane are (, 7, 0) the vector along that normal is i ˆ 7j ˆ.
7.4 D Geometry Sol : 7x + y z Direction ratios of vector normal to the plane are 7i + j k 0 (7,, -) Sol : Direction ratios of line <4, -5, 7> Direction ratio of line perpendicular to plane <, 0, > 4 + ( 5) (0) + 7 ( ) sinθ 6 + 5 + 49 9 + 0 + 4 90 Sol : x + y + z + 7 0 Distance from origin is 0 + 0 + (0) + 7 + + 9 Sol 4: x y + z 0 Distance from (,, ) is () () + () 9+ 9+ 9 Sol 5: x y + 4z Intercept on x-axis (y, z 0, 0)x 4 Intercept on y-axis (z, z 0, 0)y 6 Intercept on z-axis (x, y 0, 0)z 7 Sol 6: <a, b, c> <,, > < l, m, n>,, + 9+ 4 + 9+ 4 + 9+ 4,, 4 4 4 Sol 7: Direction cosines of y-axis <0,, 0> Sol 8: x + x + Direction ratio are y y z 5 z 5,, 5 or,, 0 Sol 9: r. (i j k) ; r. (i 6j + k) 0. + ( 6).( ) + ()( ) cosθ + 4 + 4 9 + 6 + 4 θ cos + 4.7 Sol 0: r i j + k + λ (i 6j + k) and Plane r. (i + j + k). 6. +. Sinθ 9 + 6 + 4 7 Sol : l - 5m + n 0 ; 7l + 5m n 0 l 5m n 7(5m + 9n 0mn) + 5m n 0 80m + 60n 0mn 0 6m 7mn + n 0 6m 4mn mn + n 0 m (m n) n(m n) 0 n n m or m If m n, l m, if m n, l m The following ratio are +,, or,, 6 6 6 4 4 4 Sol : Line through the points x y 4 y z 6 λ x z λ 5.5 + 4. + 6.( ) cosθ 0 56 8 θ 90 Sol : x 5 4 y 4 0 equation of line λ Let foot of is ( α, β, γ) z 4 α 5 + 4λ ; β 4 ; γ 4 λ ( α ). 4 + ( β ). 0 + ( γ ). ( ) 0 (4 + 4 λ)4 ( λ ) 0 0λ+ 0 0 λ α 5+ 4 β 4 4 γ 4 5 (, 4, 5)
Mathematics 7.4 Sol 4: cos ( δθ ) l.( l +δ l) + m.(m+δ m) + n.(n+δn) l + m + n ( l +δ l) + (m +δ m) + (n +δn) neglecting δl, δm, δn l + m + n + lδ l + mm δ + nn δ ( l + m + n ) ( l + lδ l + m + lδ m+ n + lδn) ( δθ) + lδ l + mm δ + nn δ ( δθ ) ( lδ l + mδ m + nδn) l + m + n ( l +δ l) + (m +δ m) + (n +δn) (i) ( δ l) + ( δ m) + ( δ n) lδl mδm nδn (ii) ( δθ ) ( δ l) + ( δ m) + ( δm) Sol 5: Z y O y x P z (x,y,z ) l + l + l m + m + m n + n + n l + m +.n + ll + ll + mm + mm + nm + nn Similarly dot product with l and l gives as result i.e. it makes same angle with ( l,m,n ) ( l,m,n ) and ( l,m,n ) Sol 6: x + y 0 y z 0 x (i) y z 0 (ii) The line will be across (a, b, c ) (a, b,c ) ( 0) (0 ) i j k 0 i ( ) j ( ) + k() i + j + k 0 Equation of line will be x y 0 z Sol 7: x + y + 6 0, x y 4z 0 x + y + 6 + λ (x y 4z) 0 x( + λ ) + y ( λ ) + z (-4 λ ) + 6 0 6 Distance from origin (+ λ ) + (4 λ ) + ( λ) 6 + 9λ + 6λ+ 6λ + 9+λ 6λ 6 6λ + 0 λ± Planes are 4x + y 4z + 6 0 ( λ ) x + 4y + 4z + 6 0 ( λ ) Sol 8: ax + by + cz (,, ) lies on () a + b + c (,, ) lies on () +a b + c c If to x + y + z 5 a + b + c 0 a + b a b 0 a b Equation of plane is x y + z. (i) Sol 9: P ( + r cos α, 5 + r cos β, 0 + r cos γ) are coordinates of point at distance r from (, 5, 0) along αβγ,, Point P lies on the given plane x y + z 5 + r cos α + 5 r cos β + r cos γ 0 5 r cos α r cos β + r cos γ r. units 4 + Sol 0: ax + by + cz (,, 4) a + b 4c This plane is parallel r i + j + 4k + λ (i + j + 6k) r i j + 5k + λ (i + j k) (i)
7.44 D Geometry a + b + 6c 0 a + b c 0 b 8c a 9c 9c 6c 4 c c + 8 9, b,a Equation of plane is 9x + 8y z or r. ( 9i + 8j k) Sol : al + bm + cn 0 (i) and f mn + gnl + hlm 0... (ii) f g h + + 0 l m n Comparing (i) and (iii) a b c l m n λ f g h f f l l a λ ± a λ Similarly g g m m b λ ± h λ h h n n c λ ± c λ Since, lines are cos θ l l + mm + nn 0 f g h λ λ λ 0 a b c f g h λ + + 0 a b c Sol : x x b' a' b a y z d' c' y z d c f g h + + 0 a b c These are perpendicular if aa + cc + 0 Sol : 4x y + z 0 + λ (x + y z 4) 0 x(4 +λ ) + y( +λ ) + z( λ ) 0 + 4λ (4 +λ ) + ( λ ) + ( λ ) 0 8 + + λ 0 λ 4 5y + 5z 6, equation of plane... (iii) (i) (ii) Distance from (,, ) 5+ 5+ 6 6 5 + 5 5 5 Sol 4: Ratios of line perpendicular to plane is {( 0), (5 0), (7 0)} Equation of plane is x + 5y + 7z k (, 5, 7) lies on the plane. + 5.5 + 7.7 k 78 x + 5y + 7z 78 Sol 5: Direction ratios of line to the given planes x + y y ; 5x 4y + z 5 i j k i( ) j ( + 5) + k ( 0) 5 4 0i 8j k Plane will be 0x + 8y + z k Passes through (,, ). () 8 k K +6 0x + 8y + z 6 0 5x + 9y + z 8 0 Sol 6: l+ m+ n 0 and l + m n n ( l+ m) ( ) l + m l+ m m.n 0 m 0 or n 0 l + m l + m + m.n ( l,m,n ),0, or,,0 π Angle 4 Sol 7: i j k 0 0 + 9 x 5 y + 7 i() j ( ) + k (9) i + j + 9k z + 9 Sol 8: ax + by + cz a + b + c a + c b
Mathematics 7.45 a b + c b 0 7a + b 5 +5a b 6 + a/,a,c 4 x + 4z ratio [-, 0, 4] xz plane ratio [0,, 0].0 + 0. + 4.0 0 Hence given plane is perpendicular to xz plane. Sol 9: α β ( + 8 + 5) 4 + 4 + 6 α, γ 6 6 Length + 8 + 5 4 4 (,,) γ 4 (a,b,g) x 5 β + Sol 40: ax + by + cz (, 0, ) a c 4 (,, ) a + b + c It is parallel to,, a b + c 0 4a + 5c 4 + 4c + 5c c a +b b 6 y Eq. of plane x z 4x y z 6 r (4i ˆ ˆj k) ˆ 6 Sol 4: distance between x y + z 4 x y + z 4 + Distance, d 5 5 4 4+ + 9 4 4 Sol 4: ax + by + cz A,0,0, B 0,,0, C 0,0, a b c a α, b β, c γ + + α β γ Sol 4: x y + z + 4 +λ(x y + z + ) 0 Through origin λ+ 4 0; 4 λ 4 4 4 x + y + + z 0 x y 5z r + 0 x + y 5z 0 Sol 44: x + y z 6 +λ (x + y z 8) 0 x( + λ ) + y( + λ ) + z( λ) 6 8λ 0 equation of plane xz plane 0,,0 any point on the line is ( α,,α ) Direction ratios of line i j k i + k,0, i( + ) j( + ) + k(6 4) This is parallel to plane y 0 as (, 0, ). (0,, 0) 0 α 0 i.e. (0,, ) Sol 45: The equation of Plane ax + by + cz + d +λ ( a'x + b'y + c'z + d' ) 0 (i) ( a+λ a'x ) + ( b+λ by ) + ( c+λ c'z ) + d+λ d' 0 Which parallel to line l m n
7.46 D Geometry ( a+λ a'l ) + ( b+λ b'm ) + ( c+λ c'n ) 0 al + bm + cn λ a'l+ b'm+ c'n Substituting in (i) al + bm + cn ax + by + cz + d a'x + b'y + c'z + d 0 a'l+ b'm+ c'n ( ) ( ) Sol 46: ax + by + cz b 5, c 7{given intercepts} <a, b, c> <, 0, 0> 0 a 0 y z + ; 7y + 5z 5 5 7 Sol 47: ax + by + cz a + b + c P (i) Sol 50: r i j +λ (i+ j+ k) + µ(4i j+ k) B i j k 4 Plane pass through (,, 0) i(5) j( ) + k( 4) 5i + j 6 k Equation of plane r (5i + j 6k) z 5() + ( ) 6 (0) z 4 The equation of plane r (5i + j 6k) 4 Exercise Single Correct Choice Type Sol : (C) l + m + n cos α+ cos β+ cos γ Sol : (A) ax + by + cz + d 0 to intersect x and y axis at equal angle tan α tan β a b A,0,0 ; B 0,,0 ; a b x a, y b, c z C 0,0, c Sol : (D) Parallel to x-axis i.e. <, 0, 0> x a y b 0 z c 0 P a b c + + + + from (i) Sol 48: i j + k from 5x + y 7z + 9 0 5 + 9 9 49 + 4 + 5 78 Sol 4: (C) cosα l + m + n m + + 4 m cosβ cos γ cos60 m 4 (i +λ j + k) from 5x + y 7z + 9 0 5 + λ + 9 + λ 49 + 4 + 5 78 λ or 6 + λ 9 Sol 49: Normal to vector i + 5j 6k x +5y 6z k at 7 units from origin k 6 + 5 + 9 7 ; k ± 7 70 r (i ˆ 5ˆj 6k) ˆ ± 7 70 β 60 Sol 5: (A) x+ y+ z+ λ ( +λ, + λ, λ) ( +λ ) + 4 ( λ ) + 5( λ ) 5 5λ 0 5 5λ 0 x 6 (5, 5, 4) Sol 6: (B) x 0 y z 4 (0 6 + 4 0) 9
Mathematics 7.47 x y z 4 4 4 74 44 6 96 44 6 5 6 8 4 x 8, y 8, z 4 + 4 8 8,, 6 Sol 7: (B) x + y z 4 + λ (x + 5z 4) 4 + λ λ Sol : (D) x + y or x y z + z + 4 x x + y z 5 z 4 + 4 0 x + y 8z 8 0 Sol 8: (D) i j 5k i j k It passes through (, 0, 0) Equation of line is x y z 5 i( ) j() + k( ) x y z 4 Sol 9: (C) Line is parallel to plane ax + by + cz If a + b + c 0 Only C satisfies the condition Sol 0: (A) a 49 + + 5 ; b + 0+ ;c 6 + + 0 7 + 5 + 7 s s(s a) (s b) (s c) 5 + 7 + + 7 5 + 5 7 7 + 5 7 + + 74 5 (7 + 74 6 74 + 74 6 Sol : (B) Let P(x, y, z) be any point on the locus, then the distances from the six faces are x +, x, y +, y, z +, z According to the given condition x + + x + y + + y + z + + z 0 (x + y + z ) 0 6 4 x + y + z Sol : (B) If ( x,y,z ), ( x,y,z ),( x,y,z ) and ( x 4,y 4,z4) are coplanar, then x x y y z z x x y y z z 0 x x y y z z 4 4 4 4 0 4+ 4+ 4 0 5+ + 0 9+ 4+ 4 5 5 4 6 0 0 5 4( 0 0) 5( 0 6) + 5( 40 8) 40 70 + 0 0 Sol 4: (D) The plane y+ z+ 0 Since the plane does not have any intercepts on x-axis, therefore it is parallel to x-axis. Then normal to plane can not be parallel to x-axis. Sol 5: (A) Using the fact that reflection of a' x + b' y + c'z + d' 0 in the plane ax+ by + cz + d 0 is given by (aa' + bb' + cc') (ax+ by + cz + d) (a + b + c ) (a' x + b' y + c' z + d') We get the required equation as ( + + 4) (x y + z ) ( + + )(x y + 4z ) 6 (x y + z ) x y + 4z 4x y + z 5 0
7.48 D Geometry Previous Years Questions Sol : (A) Given equation of straight line x 4 y z k Since, the line lies in the plane x 4y + z 7 Point (4,, k) must satisfy the plane. 8 8 + k 7 k 7 Sol : (B) Since, the lines intersect they must have a point in common x y+ z i.e., λ 4 x y k z and µ x λ+,y λ,z 4λ+ and x µ+, y µ+ k,z µ are same λ+ µ+, λ µ+ k, 4λ+ µ On solving Ist and IIIrd terms, we get, λ and µ 5 9 k λ µ k ( 5) 9 k Sol : (A) Since, + + a b c cuts the coordinate axes at A(a, 0, 0), B(0, b, 0), C(0, 0, c) and its distance from origin + + a b c y C z (0, 0, c) B (0, b, 0) A x (a, 0, 0) or + + (i) a b c where P is centroid of triangle a b c x,y,z (ii) From Eqs. (i) and (ii), we get + + or 9x 9y 9z + + 9 K Sol 4: Area of ABC (AB AC), where AB ˆi + ˆj kˆ and AC i ˆ + 0ˆj + 0kˆ ˆi ˆj kˆ AB AC ( ˆj k) ˆ 0 0 Area of triangle (AB AC).. 9 + 4 sq. units Sol 5: A unit vector perpendicular to the plane (PQ PR) determined by P, Q, R ± PQ PR where PQ [i ˆ + ˆj] kˆ and PR ˆi + j ˆ kˆ ˆi ˆj kˆ PQ PR ˆi( + 9) ˆj( ) + k( ˆ + ) 8i ˆ+ 4ˆj + 4kˆ PQ PR 4 4+ + 4 6 (PQ PR) 4(i ˆ+ ˆj + k) ˆ (i ˆ+ ˆj + k) ˆ Unit vector ± ± ± 4 6 6 PQ PR Sol 6: Let the equation of plane through (,, ) having a, b, c as DR s of normal to plane, a(x ) + b(y ) + c(z ) 0 and plane is parallel to straight line having DR s. (, 0, ) and (,, 0) a c 0 and a + b 0 a b c Equation of plane is x + y + z 0 or + +. Its intercept on coordinate axes are A(, 0, 0), B(0,, 0), C(0, 0, ) Hence, the volume of tetrahedron OABC 0 0 7 9 [abc] 0 0 cu units 6 6 6 0 0 K 9
Mathematics 7.49 Sol 7: Equation of plane containing the lines x y + z 0 and x + y + z 5 is (x y + z ) +λ (x + y + z 5) 0 ( + λ )x + ( λ )y + ( λ+ )z 5λ 0 Since, distance of plane from (,, ) to above plane is / 6. 6λ+ 4+λ λ 5λ (λ+ ) + ( λ ) + ( λ+ ) 6 6( λ ) λ + λ+ 6 4 λ 0, 5 Equations of planes are x y + z 0 and 6x + 9y + 9z 05 0 x y z 4 Sol 8: (C) The line + + lies in the plane, then point (,, 4) lies on the plane m 5 (i) And line is to normal of plane m (ii) From (i) and (ii) and m + m + ( ) Sol 9: (A) The eq of line passes through (, 5, 9) along x y z is x y+ 5 z 9 The point on line ( r +, r 5, r + 9) This point also lies on the given plane r+ r+ 5+ r+ 9 5 r 0 The point in ( 9, 5, ) Distance between (, 5,9) and ( 9, 5, ) ( ) ( ) r 0 + 0 + 0 0 unit Sol 0: (C) x y+ z r 4 The point of interstation ( r +, 4r, r + ) Lies on plane, then r + 4r + + r + 6 0 r 0 r 5,,4 The point in ( ) Distance ( 5 ) + ( 0) + ( 4 ) 6 + 9 + 44 69 Sol : (B) Let the two lines in a same plane interest at P x,y,0, then x 5y and x + y 5 ( ) On solving, we get P ( 4,,0) Any plane to x + y + 6z is x + y + 6z λ P( 4,,0 ) must satisfies it, then 4+ + 0 λ λ 7 The eq. to required plane x + y + 6z 7 Sol : (B) The parallel planes x + y + z 8 and 4x + y + 4z 5 8 5 Distance 6 + 4 + 6 6 7 6 Sol : (A) Image of point (,, 4) is x y z 4 ( + 4+ ) 4+ + (, 5, ) Since line is parallel to plane direction, ratio will not change x+ y 5 z Eq. of imaged line Sol 4: (A) + m+ n 0 n + m Substituting in ( ) ( m) m + n m + + m + + m + m m + m 0 m( m + ) 0 m 0,
7.50 D Geometry if m 0,, n if m, 0 n (not possible) Therefore direction cosine,0, or,,0 cosφ + ( 0) + ( 0) π φ Sol 5: (B) The lines x y z 4 and k x y 4 z 5 are coplanars, then k k k 0 ( ) kk+ 0 K 0, Two values exist. Sol 6: (A) Eq. of plane parallel to x y + z 5 0 is x y + z λ distance from origin is, then 0 0 + 0 λ λ + 4+ 4 λ± Eq. of plane x y + z ± Sol 7: (D) sin θ 5+ λ 4 5 +λ Given cosθ 5 4 + 4+ λ + 4+ 9 + 4+λ 5 sinθ cos θ 4 4 From (i) 5+ λ 4 4 5 +λ 5+λ ( 5+ λ ) ( ) 0λ 0 λ 9 5 +λ 5 + 9λ + 0λ... (i) Sol 8: (B) Statement-I: Since mid point of A(, 0, 7) and B(, 6, ) is which lies on the line, therefore point B is image of A about line Statement-II: Since it given that the line only bisects the line joining A and B, therefore not the correct explanation.,0, or,,0 cosθ + 0 + 0 π 9 ( ) ( ) JEE Advanced/Boards Exercise Sol : Let point P be taken as origin and q,s are the position vectors of Q and S points respectively. PR q + s s P (o) Y 4 z q + 4q ( + s ) 5q + 4s P.V. of X 5 5 4s + q + s q + 5s P.V. of Y 5 5 Let, PZ ZR YZ and λ ZX µ 4 q µ q+ s + + s q+ s 5 5 P.V. of P λ+ µ+ µ+ 5 λ+ µ+ µ+ 5 λ+ µ+ R X 4 Q (i) (ii)
Mathematics 7.5 From (i) & (ii), we get 4 µ 4, λ PZ ZR 4 PZ PR 5 Sol : p (x,y) PA PB + OA OB 0 (x )(x + ) + y + ( ) 0 x + y 4 PA PB (x ) + y (5 x) (5 + x) Max is Min 9 M + m 4 5 5 M (x + ) + y 5 4x Sol : a b c a+ b a + b + a b ; ab θ 0 a + b & b ( ) (a + b) (b) a + b b cos θ a. b + b + cosθ k cosθ 0 θ π Sol 4: c λ a + µb c λ + µ + λ µa. b λ + µ λ µ c.a λ + µ (a. b) 0 µ λ 0 µ λ u λ λ + 4λ λ 4a + b + 4a. b b cosθ λ, µ, Sol 5: P xa+ yc p yc 0 p.a x x [0,] 0 p.b xa.b x [,0] x 0 p.c y a b c + p.a p.b p.c + a + b p xa + y y y Sol 6: For max. x and y; x + y x x ; y p a + b Sol 7: The coordinates of the points, O and P, are (0, 0, 0) and (,, -) respectively. Therefore, the direction ratios of OP are ( - 0), ( - 0) and (--0) - It is knows that the equation of the plane passing through the point ( x yz ) is ( ) ( ) ( ) ax x + by y + cz z 0, where a, b and c are the direction ratio of normal. Here, the direction ratios of normal are, and - and the point P is (,, -). Thus, the equation of the required plane is x ( ) + y ( ) z ( + ) 0 x + y z 4 0 Sol 8: (i) A [xyz ] ; B [xyz ] ; C [xyz ] AB ( C) 0all are coplanar A B 0 B C C A i.e. all are mutually which simultaneously is not possible. (ii) P (x,y,x ) Q (yyy )O (0,0,0) In POQ; OP xi+ xj+ xk OQ yi+ yj+ yk OP. OQ xy + xy + xy x > 0 y > 0 OP. OQ < 0 [i.e. it can never be zero]
7.5 D Geometry Sol 9: A ( 5,,5) ; B (,,) ; C (4,,) D (,, ) and AEF DE DA + DB S OE OD OA OD + OB OD OE 5i+ j+ 5k + i+ j+ k ( i+ j k) OE i + j + k BF BA + BC OF OA + OC OB OF 5i + j + 5k + 4i + j + k i j k OF i + j + 4k Area AE AF ˆi ˆj kˆ 0 6 (i + 6k) (i + j k) ˆ ˆ ˆ ˆ ˆ ˆ i( 6) j( 8) k() 6i 0 j k + + + 6 + 400 + 4 440 4 Sol 0: ( ) ( (a ) β + (b ) β+ c) y + (a ) α + (b ) α+ c x + ( )( ) (a ) γ + (b ) γ+ c x y 0 a b c 0 a ; b ; c 0 a + b + c 0 S 0 Sol : The equation of the given line is r i ˆ ˆj + k ˆ +λ (i ˆ + 4ˆj + k) ˆ The equation of the given plane is ( ˆ ˆ ˆ) r. i j+ k 5 (i) (ii) Substituting the value of from equation (i) in equation (ii), we obtain. + +λ ( + + ) ( + ) ( ) ˆ ( ) ˆ( ) ˆ ( ˆ ˆ ˆ) i ˆ ˆj kˆ i ˆ 4ˆj kˆ ˆi ˆj kˆ 5 λ+ i+ 4λ jλ+ k.i j+ k 5 ( λ+ ) ( 4λ ) + ( λ+ ) 5 λ0 Substituting this value in equation (i), we obtain the equation of the line as r i ˆ ˆj + kˆ This means that the position vector of the point of intersection of the line and the plane is r i ˆ ˆj + kˆ This shows that the point of intersection of the given line and plane is given by the coordinates (, -, ). The point is (-, -5, -0). The distance d between the points, (, -, ) and (-, -5, -0), is ( ) ( ) ( ) d + 5+ + 0 9 + 6 + 44 69 x y z Sol : a b c Parallel to the plane x + 5y + 4z 0 a + 5b + 4c 0 ( + λ, +λ, 4+ λ ) ( + ka, + kb, + kc) ka + kb 7 + kc b a 7 5 b c a c 0b 7a c..(i) 7a 0b a 5b 4 a a b ; c a b c Sol : λ a b c λ a + k ; λ b + k; λ c k a + b + c 6 a + b + c + k + k k a b c a b c c b a b c a c (a + b + c ) (a + b c) a, b, c or a b, c
Mathematics 7.5 Exercise Single Correct Choice Type Sol : (B) Direction cosines of PQ (,, 6) (, 4, 5) Ratios, + 4, 6 5, 7, Direction cosines, 7, 7 7 7 or, 7, 7 7 7 m+ 4m + Sol : (A) α, β, m+ m+ As α 0 m [A] 6m 5 γ m+ Sol : (D) x + y + z + 5 x + y z x y λ z 7x + 0y 8z are to each other i j k. i( 5) j ( 6 ) + k( ) 5i + 7j + k. i j k λ 7 + 0 8 i (8 + 0 λ ) j ( 6 + 7λ ) + k (+0 + 7) 40 50 λ + 49 λ 7 0 λ Sol 4: (A) m (,4,5) ( a,b,g ) (,5,7) m + α 0 m m+ Sol 5: (B) cosα cos α+ cos β+ cos γ+ cos δ + + + 4 Sol 6: (A) α(x a) +β(y b) +γ(z c) 0 β A b +γ c +α a,0,0 α α a+β b+γc α a+β b+γc α a+β b+γc (h,k, l ),, α β γ (h a) α β b+γc (k b) β α a+γc ( l c) γ α a+βb h a b c a k b c 0 a b l c (h α ) (k b)( l c) bc + b al + ac ac c [ab + ak ab] 0 (h α ) [kl kc b l ] bal cak 0 hkl hkc hbl akl + akc + abl bal cak 0 ayz + bzx + cxy xyz Sol 7: (A) x y + z + pp + qq + rr 6 + + > 0 +ve acute x y + z + 45 0[C] Sol 8: (C) (x y + 6z + 8) 7 x 4/ y+ 6/5 z / λ 4 4 6 + λ, + λ, + 4λ 5 5 8 9 4 6,, + 5 λ, + 8 λ, + 9λ 5 5 + 4 6 Both passes through,, 5 Minimum distance is zero. Sol 9: (C) + β γ α (b + c) +β (a + c) +γ (a + b) 0 α(b c) +β(c a) +γ(a b) 0 α b+β c+γ a 0 ; α c+β a+γ b 0 α a β b γ c bc ac ab x a bc b γ ac c Assertion Reasoning Type z ab Sol 0: (C) y + z + 0 [0,, ] x-axis [, 0, 0] sinθ 0 R is wrong.