QUESTION BANK ON STRAIGHT LINE AND CIRCLE


 Mark Sherman
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1 QUESTION BANK ON STRAIGHT LINE AND CIRCLE
2 Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =, β = (B) α =, β = ± (C) α =, β = ± (D) α = ±, β = Q. The axes are translated so that the new equation of the circle x² + y² 5x + y 5 = 0 has no first degree terms. Then the new equation is : x + y = 9 (B) x + y = 49 (C) x + y = of these Q. Given the family of lines, a (x + 4y + 6) + b (x + y + ) = 0. The line of the family situated at the greatest distance from the point P (, ) has equation : 4x + y + 8 = 0 (B) 5x + y + 0 = 0 (C) 5x + 8y + 0 = 0 Q.4 The ends of a quadrant of a circle have the coordinates (, ) and (, ) then the centre of the such a circle is (, ) (B) (, ) (C) (, 6) (D) (4, 4) Q.5 The straight line, ax + by = makes with the curve px + a xy + qy = r a chord which subtends a right angle at the origin. Then : r (a + b ) = p + q (B) r (a + p ) = q + b (C) r (b + q ) = p + a Q.6 The circle described on the line joining the points (0, ), (a, b) as diameter cuts the x axis in points whose abscissae are roots of the equation : x² + ax + b = 0 (B) x² ax + b = 0 (C) x² + ax b = 0 (D) x² ax b = 0 Q.7 Centroid of the triangle, the equations of whose sides are x 0xy + 7y = 0 and x y + 4=0 is 8 8 (, ) (B), 8 8 (C), (D), Q.8 The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is 5 (B) 5 (C) 5 (D) 5 Q.9 The line x + y = 0 bisects the angle between a pair of straight lines of which one has equation x 7y + 5 = 0. The equation of the other line is : x + y = 0 (B) x y + = 0 (C) 5x + 5y = 0 Q.0 Given two circles x² + y² 6x y + 5 = 0 & x² + y² + 6x + y + 5 = 0. The tangent at (, ) to the first circle : passes outside the second circle (B) touches the second circle (C) intersects the second circle in real points (D) passes through the centre of the second circle. Q. A variable rectangle PQRS has its sides parallel to fixed directions. Q & S lie respectively on the lines x = a, x = a & P lies on the x axis. Then the locus of R is : a straight line (B) a circle (C) a parabola (D) pair of straight lines Q. To which of the following circles, the line y x + = 0 is normal at the point x y 9 (B) x y (C) x² + (y )² = 9 (D) (x )² + y² = 9 9,? []
3 Q. On the portion of the straight line, x + y = 4 intercepted between the axes, a square is constructed on the side of the line away from the origin. Then the point of intersection of its diagonals has coordinates (, ) (B) (, ) (C) (, ) (D) (, ) Q.4 The locus of the mid point of a chord of the circle x² + y² = 4 which subtends a right angle at the origin is x + y = (B) x² + y² = (C) x² + y² = (D) x + y = Q.5 Given the family of lines, a (x + y + 4) + b (x y ) = 0. Among the lines of the family, the number of lines situated at a distance of 0 from the point M (, ) is : 0 (B) (C) (D) Q.6 The equation of the line passing through the points of intersection of the circles ; x² + y² x + y 9 = 0 & x² + y² + 6x + y 5 = 0 is : 0x y 8 = 0 (B) 5x + y 8 = 0 (C) 5x y 8 = 0 (D) 0x + y + = 0 Q.7 Through a point A on the xaxis a straight line is drawn parallel to yaxis so as to meet the pair of straight lines ax + hxy + by = 0 in B and C. If AB = BC then h = 4ab (B) 8h = 9ab (C) 9h = 8ab (D) 4h = ab Q.8 The number of common tangent(s) to the circles x + y + x + 8y = 0 and x + y 4x 0y + 9 = 0 is (B) (C) (D) 4 Q.9 A, B and C are points in the xy plane such that A(, ) ; B (5, 6) and AC = BC. Then ABC is a unique triangle (B) There can be only two such triangles. (C) No such triangle is possible (D) There can be infinite number of such triangles. Q.0 From the point A (0, ) on the circle x² + 4x + (y )² = 0 a chord AB is drawn & extended to a point M such that AM = AB. The equation of the locus of M is : x² + 8x + y² = 0 (B) x² + 8x + (y )² = 0 (C) (x )² + 8x + y² = 0 (D) x² + 8x + 8y² = 0 Q. If A (, p ) ; B (0, ) and C (p, 0) are the coordinates of three points then the value of p for which the area of the triangle ABC is minimum, is (B) (C) or Q. The area of the quadrilateral formed by the tangents from the point (4, 5) to the circle x² + y² 4x y = 0 with the pair of radii through the points of contact of the tangents is : 4 sq.units (B) 8 sq.units (C) 6 sq.units Q. The area of triangle formed by the lines x + y = 0, x y + 9 = 0 and x y + = sq. units (B) sq. units (C) 4 sq. units (D) 9 sq. units 7 7 Q.4 Two circles of radii 4 cms & cm touch each other externally and θ is the angle contained by their direct common tangents. Then sin θ = 4 5 (B) 5 (C) 4 Q.5 The set of lines ax + by + c = 0, where a + b + 4c = 0, is concurrent at the point :, (B), (C), (D) (, ) []
4 Q.6 The locus of poles whose polar with respect to x² + y² = a² always passes through (K, 0) is Kx a² = 0 (B) Kx + a² = 0 (C) Ky + a² = 0 (D) Ky a² = 0 Q.7 The co ordinates of the point of reflection of the origin (0, 0) in the line 4x y 5 = 0 is : (, ) (B) (, ) (C) 4, (D) (, 5) 5 5 Q.8 The locus of the mid points of the chords of the circle x + y ax by = 0 which subtend a right angle a b at, is ax + by = 0 (B) ax + by = a + b (C) x + y ax by + a b 8 = 0 (D) x + y ax by Q.9 A ray of light passing through the point A (, ) is reflected at a point B on the x axis and then passes through (5, ). Then the equation of AB is : 5x + 4y = (B) 5x 4y = (C) 4x + 5y = 4 (D) 4x 5y = 6 Q.0 From (, 4) chords are drawn to the circle x² + y² 4x = 0. The locus of the mid points of the chords is x² + y² 5x 4y + 6 = 0 (B) x² + y² + 5x 4y + 6 = 0 (C) x² + y² 5x + 4y + 6 = 0 (D) x² + y² 5x 4y 6 = 0 Q. m, n are integer with 0 < n < m. A is the point (m, n) on the cartesian plane. B is the reflection of A in the line y = x. C is the reflection of B in the yaxis, D is the reflection of C in the xaxis and E is the reflection of D in the yaxis. The area of the pentagon ABCDE is m(m + n) (B) m(m + n) (C) m(m + n) (D) m(m + n) Q. Which one of the following is false? The circles x² + y² 6x 6y + 9 = 0 & x² + y² + 6x + 6y + 9 = 0 are such that : they do not intersect (B) they touch each other (C) their exterior common tangents are parallel (D) their interior common tangents are perpendicular. Q. The lines y y = m (x x ) ± a m are tangents to the same circle. The radius of the circle is a/ (B) a (C) a Q.4 The centre of the smallest circle touching the circles x² + y² y = 0 and x² + y² 8x 8y + 9 = 0 is : (, ) (B) (4, 4) (C) (, 7) (D) (, 5) Q.5 The ends of the base of an isosceles triangle are at (, 0) and (0, ) and the equation of one side is x = then the orthocentre of the triangle is 5 4 7, (B), (C), (D), Q.6 A rhombus is inscribed in the region common to the two circles x + y 4x = 0 and x + y + 4x = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is 8 sq.units (B) 4 sq.units (C) 6 sq.units a b 8 = 0 [4]
5 Q.7 A variable straight line passes through a fixed point (a, b) intersecting the co ordinates axes at A & B. If 'O' is the origin then the locus of the centroid of the triangle OAB is : bx + ay xy = 0 (B) bx + ay xy = 0 (C) ax + by xy = 0 Q.8 The angle between the two tangents from the origin to the circle (x 7) + (y + ) = 5 equals 4 (B) (C) Q.9 If P = (, 0); Q = (, 0) & R = (, 0) are three given points, then the locus of the points S satisfying the relation, SQ + SR = SP is : a straight line parallel to x axis (B) a circle passing through the origin (C) a circle with the centre at the origin (D) a straight line parallel to y axis. Q.40 The equation of the circle having normal at (, ) as the straight line y = x and passing through the point (, ) is : x² + y² 5x + 5y + = 0 (B) x² + y² + 5x 5y + = 0 (C) x² + y² 5x 5y = 0 (D) x² + y² 5x 5y + = 0 Q.4 The equation of the base of an equilateral triangle ABC is x + y = and the vertex is (, ). The area of the triangle ABC is : 6 (B) 6 (C) 8 Q.4 In a right triangle ABC, right angled at A, on the leg AC as diameter, a semicircle is described. The chord joining A with the point of intersection D of the hypotenuse and the semicircle, then the length AC equals to AB AD AB AD (B) AB AD AB AD (C) AB AD (D) AB AD AB AD Q.4 The equation of the pair of bisectors of the angles between two straight lines is, x 7xy y = 0. If the equation of one line is y x = 0 then the equation of the other line is : 4x 8y = 0 (B) 8x 4y = 0 (C) 8x + 4y = 0 (D) 4x + 8y = 0 Q.44 If the circle C : x + y = 6 intersects another circle C of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to /4, then the coordinates of the centre of C are : 9, (B) , 5 (C) 9, (D) 5 5 Q.45 Area of the rhombus bounded by the four lines, ax ± by ± c = 0 is : c ab (B) c ab (C) 4 c ab (D) 5 ab 4 c 9, 5 Q.46 Two lines p x + q y + r = 0 & p x + q y + r = 0 are conjugate lines w.r.t. the circle x² + y² = a² if p p + q q = r r (B) p p + q q + r r = 0 (C) a²(p p + q q ) = r r (D) p p + q q = a² r r Q.47 Area of the quadrilateral formed by the lines x + y = is : 8 (B) 6 (C) 4 Q.48 If the two circles (x )² + (y )² = r² & x² + y² 8x + y + 8 = 0 intersect in two distinct points then < r < 8 (B) r < (C) r = (4) r > [5]
6 Q.49 Let the algebraic sum of the perpendicular distances from the points (, 0), (0, ) & (, ) to a variable straight line be zero, then the line passes through a fixed point whose coordinates are : 5 5 (, ) (B) (, ) (C), (D), 5 5 Q.50 If a circle passes through the point (a, b) & cuts the circle x² + y² = K² orthogonally, then the equation of the locus of its centre is : ax + by (a² + b² + K²) = 0 (B) ax + by (a² b² + K²) = 0 (C) x² + y² ax 4by + (a² + b² K²) = 0 (D) x² + y² ax by + (a² b² K²) = 0 Q.5 Consider a quadratic equation in Z with parameters x and y as Z xz + (x y) = 0 The parameters x and y are the coordinates of a variable point P w.r.t. an orthonormal coordinate system in a plane. If the quadratic equation has equal roots then the locus of P is a circle (B) a line pair through the origin of coordinates with slope / and / (C) a line pair through the origin of coordinates with slope / and (D) a line pair through the origin of coordinates with slope / and / Q.5 Consider the circle S x + y 4x 4y + 4 = 0. If another circle of radius 'r' less than the radius of the circle S is drawn, touching the circle S, and the coordinate axes, then the value of 'r' is (B) 4 (C) 7 4 (D) 6 4 Q.5 Vertices of a parallelogram ABCD are A(, ), B(, 6), C(, ) and D(, 6). If a line passing through the origin divides the parallelogram into two congruent parts then the slope of the line is 5 (B) (C) (D) Q.54 The distance between the chords of contact of tangents to the circle ; x + y + gx + fy + c = 0 from the origin & the point (g, f) is : g f (B) g f c (C) g f c g f (D) g f c g f Q.55 Two mutually perpendicular straight lines through the origin from an isosceles triangle with the line x + y = 5. Then the area of the triangle is 5 (B) (C) 5/ (D) Q.56 The locus of the centers of the circles which cut the circles x + y + 4x 6y + 9 = 0 and x + y 5x + 4y = 0 orthogonally is 9x + 0y 7 = 0 (B) x y + = 0 (C) 9x 0y + =0 (D) 9x + 0y + 7 = 0 Q.57 Distance between the two lines represented by the line pair, x 4xy + 4y + x y 6 = 0 is : (B) 5 (C) 5 5 Q.58 The locus of the center of the circles such that the point (, ) is the mid point of the chord 5x + y = 6 is : x 5y + = 0 (B) x + 5y = 0 (C) x + 5y + = 0 [6]
7 Q.59 The distance between the two parallel lines is unit. A point 'A' is chosen to lie between the lines at a distance 'd' from one of them. Triangle ABC is equilateral with B on one line and C on the other parallel line. The length of the side of the equilateral triangle is d d (B) d d (C) d d (D) d d Q.60 The locus of the mid points of the chords of the circle x² + y² + 4x 6y = 0 which subtend an angle of radians at its circumference is : (x )² + (y + )² = 6.5 (B) (x + )² + (y )² = 6.5 (C) (x + )² + (y )² = 8.75 (D) (x + )² + (y + )² = 8.75 Q.6 Given A(0, 0) and B(x, y) with x (0, ) and y > 0. Let the slope of the line AB equals m. Point C lies on the line x = such that the slope of BC equals m where 0 < m < m. If the area of the triangle ABC can be expressed as (m m ) f (x), then the largest possible value of f (x) is (B) / (C) /4 (D) /8 Q.6 If two chords of the circle x + y ax by = 0, drawn from the point (a, b) is divided by the x axis in the ratio : then: a > b (B) a < b (C) a > 4 b (D) a < 4 b Q.6 P lies on the line y = x and Q lies on y = x. The equation for the locus of the mid point of PQ, if PQ = 4, is 5x + 6xy + y = 4 (B) 5x 6xy + y = 4 (C) 5x 6xy y = 4 (D) 5x + 6xy y = 4 Q.64 The points (x, y ), (x, y ), (x, y ) & (x, y ) are always : collinear (B) concyclic (C) vertices of a square (D) vertices of a rhombus Q.65 If the vertices P and Q of a triangle PQR are given by (, 5) and (4, ) respectively, and the point R moves along the line N: 9x + 7y + 4 = 0, then the locus of the centroid of the triangle PQR is a straight line parallel to PQ (B) QR (C) RP (D) N Q.66 The angle at which the circles (x ) + y = 0 and x + (y ) = 5 intersect is (B) (C) (D) 6 4 Q.67 The co ordinates of the points A, B, C are ( 4, 0), (0, ) & (, ) respectively. The point of intersection of the line which bisects the angle CAB internally and the line joining C to the middle point of AB is 7 4, (B) 5, (C) 7 0, (D) 5, Q.68 Two congruent circles with centres at (, ) and (5, 6) which intersect at right angles has radius equal to (B) (C) 4 Q.69 Three lines x + y + = 0 ; x + y 7 = 0 and x y 4 = 0 form the three sides of two squares. The equation to the fourth side of each square is x y + 4 = 0 & x y + 6 = 0 (B) x y + 4 = 0 & x y 6 = 0 (C) x y 4 = 0 & x y 6 = 0 (D) x y 4 = 0 & x y + 6 = 0 [7]
8 Q.70 A circle of radius unity is centred at origin. Two particles start moving at the same time from the point (, 0) and move around the circle in opposite direction. One of the particle moves counterclockwise with constant speed v and the other moves clockwise with constant speed v. After leaving (, 0), the two particles meet first at a point P, and continue until they meet next at point Q. The coordinates of the point Q are (, 0) (B) (0, ) (C) (0, ) (D) (, 0) Q.7 The points A(a, 0), B(0, b), C(c, 0) & D(0, d) are such that ac = bd & a, b, c, d are all non zero. The the points form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic Q.7 The value of 'c' for which the set, {(x, y) x + y + x } {(x, y) x y + c 0} contains only one point in common is : (, ] [, ) (B) {, } (C) { } (D) { } Q.7 Given A (, ) and AB is any line through it cutting the xaxis in B. If AC is perpendicular to AB and meets the yaxis in C, then the equation of locus of mid point P of BC is x + y = (B) x + y = (C) x + y = xy (D) x + y = Q.74 A circle is inscribed into a rhombous ABCD with one angle 60º. The distance from the centre of the circle to the nearest vertex is equal to. If P is any point of the circle, then PA PB PC PD is equal to : (B) (C) 9 Q.75 The number of possible straight lines, passing through (, ) and forming a triangle with coordinate axes, whose area is sq. units, is one (B) two (C) three (D) four Q.76 P is a point (a, b) in the first quadrant. If the two circles which pass through P and touch both the coordinate axes cut at right angles, then : a 6ab + b = 0 (B) a + ab b = 0 (C) a 4ab + b = 0 (D) a 8ab + b = 0 Q.77 In a triangle ABC, if A (, ) and 7x 0y + = 0 and x y + 5 = 0 are equations of an altitude and an angle bisector respectively drawn from B, then equation of BC is x + y + = 0 (B) 5x + y + 7 = 0 (C) 4x + 9y + 0 = 0 (D) x 5y 7 = 0 Q.78 The range of values of 'a' such that the angle θ between the pair of tangents drawn from the point (a, 0) to the circle x + y = satisfies < θ < π is : (, ) (B), (C), (D),, Q.79 Distance of the point (, 5) from the line x + y + 4 = 0 measured parallel to the line x 4y + 8 = 0 is 5/ (B) 9/ (C) 5 (D) None Q.80 Three concentric circles of which the biggest is x + y =, have their radii in A.P. If the line y = x + cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is 0, (B) 0, (C) 0, 4 4 [8]
9 Q.8 The coordinates of the vertices P, Q, R & S of square PQRS inscribed in the triangle ABC with vertices A (0, 0), B (, 0) & C (, ) given that two of its vertices P, Q are on the side AB are respectively ,,,,, &, (B) ,,,,, &, (C) (, 0), 0,,, &, (D) ,,,,, &, Q.8 A tangent at a point on the circle x + y = a intersects a concentric circle C at two points P and Q. The tangents to the circle X at P and Q meet at a point on the circle x + y = b then the equation of circle is x + y = ab (B) x + y = (a b) (C) x + y = (a + b) (D) x + y = a + b Q.8 AB is the diameter of a semicircle k, C is an arbitrary point on the semicircle (other than A or B) and S is the centre of the circle inscribed into triangle ABC, then measure of angle ASB changes as C moves on k. (B) angle ASB is the same for all positions of C but it cannot be determined without knowing the radius. (C) angle ASB = 5 for all C. (D) angle ASB = 50 for all C. Q.84 Tangents are drawn to the circle x + y = at the points where it is met by the circles, x + y (λ + 6) x + (8 λ) y = 0. λ being the variable. The locus of the point of intersection of these tangents is : x y + 0 = 0 (B) x + y 0 = 0 (C) x y + 0 = 0 (D) x + y 0 = 0 Q.85 Given x y = and ax + by = are two variable lines, 'a' and 'b' being the parameters connected by a b the relation a + b = ab. The locus of the point of intersection has the equation x + y + xy = 0 (B) x + y xy + = 0 (C) x + y + xy + = 0 (D) x + y xy = 0 Q.86 B & C are fixed points having co ordinates (, 0) and (, 0) respectively. If the vertical angle BAC is 90º, then the locus of the centroid of the Δ ABC has the equation : x + y = (B) x + y = (C) 9 (x + y ) = (D) 9 (x + y ) = 4 Q.87 The set of values of 'b' for which the origin and the point (, ) lie on the same side of the straight line, a x + a by + = 0 a R, b > 0 are : b (, 4) (B) b (0, ) (C) b [0, ] (D) (, ) Q.88 If a,, b,, c, & d, are four distinct points on a circle of radius 4 units then, a b c d abcd is equal to 4 (B) /4 (C) (D) 6 Q.89 Triangle formed by the lines x + y = 0, x y = 0 and lx + my =. If l and m vary subject to the condition l + m = then the locus of its circumcentre is (x y ) = x + y (B) (x + y ) = (x y ) (C) (x + y ) = 4x y (D) (x y ) = (x + y ) Q.90 Tangents are drawn to a unit circle with centre at the origin from each point on the line x + y = 4. Then the equation to the locus of the middle point of the chord of contact is (x + y ) = x + y (B) (x + y ) = x + y (C) 4 (x + y ) = x + y [9]
10 Q.9 The co ordinates of three points A( 4, 0) ; B(, ) and C(, ) determine the vertices of an equilateral trapezium ABCD. The co ordinates of the vertex D are : (6, 0) (B) (, 0) (C) ( 5, 0) (D) (9, 0) Q.9 ABCD is a square of unit area. A circle is tangent to two sides of ABCD and passes through exactly one of its vertices. The radius of the circle is (B) (C) (D) Q.9 A parallelogram has of its vertices as (, ), (, 8) and (4, ). The sum of all possible xcoordinates for the 4 th vertex is (B) 8 (C) 7 (D) 6 Q.94 A pair of tangents are drawn to a unit circle with centre at the origin and these tangents intersect at A enclosing an angle of 60. The area enclosed by these tangents and the arc of the circle is 6 (B) (C) 6 (D) 6 Q.95 The image of the pair of lines represented by ax + h xy + by = 0 by the line mirror y = 0 is ax h xy by = 0 (B) bx h xy + ay = 0 (C) bx + h xy + ay = 0 (D) ax h xy + by = 0 Q.96 A straight line with slope and yintercept 5 touches the circle, x + y + 6x + y + c = 0 at a point Q. Then the coordinates of Q are ( 6, ) (B) ( 9, ) (C) ( 0, 5) (D) ( 6, 7) Q.97 The acute angle between two straight lines passing through the point M( 6, 8) and the points in which the line segment x + y + 0 = 0 enclosed between the coordinate axes is divided in the ratio : : in the direction from the point of its intersection with the xaxis to the point of intersection with the yaxis is π/ (B) π/4 (C) π/6 (D) π/ Q.98 A variable circle cuts each of the circles x + y x = 0 & x + y 4x 5 = 0 orthogonally. The variable circle passes through two fixed points whose co ordinates are : , 0 (B), 0 (C), 0 (D), 0 Q.99 If in triangle ABC, A (, 0), circumcentre, and orthocentre 4 coordinates of midpoint of side opposite to A is : (, /) (B) (, 5) (C) (, ) (D) (, 6), then the Q.00 The radical centre of three circles taken in pairs described on the sides of a triangle ABC as diametres is the centroid of the Δ ABC (B) incentre of the Δ ABC (C) circumcentre o the Δ ABC (D) orthocentre of the Δ ABC Q.0 The line x + y = p meets the axis of x & y at A & B respectively. A triangle APQ is inscribed in the triangle OAB, O being the origin, with right angle at Q. P and Q lie respectively on OB and AB. If the area of the triangle APQ is /8 th of the area of the triangle OAB, then AQ is equal to : BQ (B) / (C) / (D) [0]
11 Q.0 Two circles are drawn through the points (, 0) and (, ) to touch the axis of y. They intersect at an angle cot 4 (B) cos 4 5 (C) (D) tan Q.0 In a triangle ABC, side AB has the equation x + y = 9 and the side AC has the equation, x + y = 6. If the mid point of BC is (5, 6) then the equation of BC is : x y = (B) 5 x y = (C) x + y = (D) x 4 y = 9 Q.04 If the line x cos θ + y sin θ = is the equation of a transverse common tangent to the circles x + y = 4 and x + y 6 x 6y + 0 = 0, then the value of θ is : 5π/6 (B) π/ (C) π/ (D) π/6 Q.05 ABC is an isosceles triangle. If the coordinates of the base are (, ) and (, 7), then coordinates of vertex A can be :, 5 (B) 8, 5 (C) 5, 5 6 (D) 7, 8 Q.06 A circle is drawn with yaxis as a tangent and its centre at the point which is the reflection of (, 4) in the line y = x. The equation of the circle is x + y 6x 8y + 6 = 0 (B) x + y 8x 6y + 6 = 0 (C) x + y 8x 6y + 9 = 0 (D) x + y 6x 8y + 9 = 0 Q.07 A is a point on either of two lines y + x = at a distance of 4 units from their point of intersection. The coordinates of the foot of perpendicular from A on the bisector of the angle between them are, (B) (0, 0) (C), (D) (0, 4) Q.08 A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of co ordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is : (x + y ) x y = 4 a (B) (x + y ) x y = a (C) (x + y ) x y = 4 a (D) (x + y ) x y = a Q.09 Three straight lines are drawn through a point P lying in the interior of the Δ ABC and parallel to its sides. The areas of the three resulting triangles with P as the vertex are s, s and s. The area of the triangle in terms of s, s and s is : ss ss ss (B) s s s (C) s s s Q.0 The circle passing through the distinct points (, t), (t, ) & (t, t) for all values of ' t ', passes through the point : (, ) (B) (, ) (C) (, ) (D) (, ) Q. The sides of a Δ ABC are x y + 5 = 0 ; x + y 5 = 0 and x y 5 = 0. Sum of the tangents of its interior angles is : 6 (B) 7/4 (C) 9 []
12 Q. If a circle of constant radius k passes through the origin 'O' and meets coordinate axes at A and B then the locus of the centroid of the triangle OAB is x + y = (k) (B) x + y = (k) (C) x + y = (4k) (D) x + y = (6k) Q. Chords of the curve 4x + y x + 4y = 0 which subtend a right angle at the origin pass through a fixed point whose coordinates are 4, (B) 4 4, (C), (D) 4, 5 5 Q.4 Let x & y be the real numbers satisfying the equation x 4x + y + = 0. If the maximum and minimum values of x + y are M & m respectively, then the numerical value of M m is : (B) 8 (C) 5 of these Q.5 If the straight lines joining the origin and the points of intersection of the curve 5x + xy 6y + 4x y + = 0 and x + ky = 0 are equally inclined to the coordinate axes then the value of k : is equal to (B) is equal to (C) is equal to (D) does not exist in the set of real numbers. Q.6 A line meets the coordinate axes in A & B. A circle is circumscribed about the triangle OAB. If d & d are the distances of the tangent to the circle at the origin O from the points A and B respectively, the diameter of the circle is : d d d d (B) d d (C) d + d (D) d d Q.7 A pair of perpendicular straight lines is drawn through the origin forming with the line x + y = 6 an isosceles triangle right angled at the origin. The equation to the line pair is : 5x 4xy 5y = 0 (B) 5x 6xy 5y = 0 (C) 5x + 4xy 5y = 0 (D) 5x + 6xy 5y = 0 Q.8 The equation of a line inclined at an angle 4 to the axis X, such that the two circles x + y = 4, x + y 0x 4y + 65 = 0 intercept equal lengths on it, is x y = 0 (B) x y + = 0 (C) x y + 6 = 0 (D) x y 6 = 0 Q.9 If the line y = mx bisects the angle between the lines ax + h xy + by = 0 then m is a root of the quadratic equation : hx + (a b) x h = 0 (B) x + h (a b) x = 0 (C) (a b) x + hx (a b) = 0 (D) (a b) x hx (a b) = 0 Q.0 Tangents are drawn from any point on the circle x + y = R to the circle x + y = r. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals r (B) r r 4r (C) (D) 5 Q. An equilateral triangle has each of its sides of length 6 cm. If (x, y ); (x, y ) & (x, y ) are its vertices then the value of the determinant, x x x y y y is equal to : 9 (B) 4 (C) 486 (D) 97 []
13 Q. A variable circle C has the equation x + y (t t + )x (t + t)y + t = 0, where t is a parameter. If the power of point P(a,b) w.r.t. the circle C is constant then the ordered pair (a, b) is 0, 0 (B), 0 0 (C) 0, 0 (D), 0 Q. Points A & B are in the first quadrant; point 'O' is the origin. If the slope of OA is, slope of OB is 7 and OA = OB, then the slope of AB is: /5 (B) /4 (C) / (D) / Q.4 Let C be a circle with two diameters intersecting at an angle of 0 degrees. A circle S is tangent to both the diameters and to C, and has radius unity. The largest radius of C is + 6 (B) + 6 (C) 6 of these Q.5 The coordinates of a point P on the line x y + 5 = 0 such that PA PB is maximum where A is (4, ) and B is (, 4) will be : (, 7) (B) (, 7) (C) (, 7) (D) (0, 5) Q.6 A straight line l with equation x y + 0 = 0 meets the circle with equation x + y = 00 at B in the first quadrant. A line through B, perpendicular to l cuts the yaxis at P (0, t). The value of 't' is (B) 5 (C) 0 (D) 5 Q.7 A variable circle C has the equation x + y (t t + )x (t + t)y + t = 0, where t is a parameter. The locus of the centre of the circle is a parabola (B) an ellipse (C) a hyperbola (D) pair of straight lines Q.8 Let a and b represent the length of a right triangle's legs. If d is the diameter of a circle inscribed into the triangle, and D is the diameter of a circle superscribed on the triangle, then d + D equals a + b (B) (a + b) (C) (a + b) (D) a b Select the correct alternatives : (More than one are correct) Q.9 The area of triangle ABC is 0 cm. The co ordinates of vertex A are ( 5, 0) and B are (, 0). The vertex C lies on the line, x y =. The co ordinates of C are (5, ) (B) (, 5) (C) ( 5, 7) (D) (7, 5) Q.0 A point (x, y ) is outside the circle, x + y + gx + fg + c = 0 with centre at the origin and AP, AQ are tangents to the circle. Then : area of the quadriletral formed by the pair of tangents and the corresponding radii through the points of contact is g f cx y gx fy c (B) equation of the circle circumscribing the ΔAPQ is, x + y + x(g x ) + y(f y ) (gx + fy ) = 0 (C) least radius of a circle passing through the points 'A' & the origin is, ( x g) ( y f) (D) the between the two tangent is, sin g f cx y gx fy c ( x g) ( y f) Q. Let u ax + by + a b = 0 v bx ay + b a = 0 a, b R be two straight lines. The equation of the bisectors of the angle formed by k u k v = 0 & k u + k v = 0 for non zero real k & k are: u = 0 (B) k u + k v = 0 (C) k u k v = 0 (D) v = 0 0 []
14 Q. x x cos = y y sin = r, represents : equation of a straight line, if θ is constant & r is variable (B) equation of a circle, if r is constant & θ is a variable (C) a straight line passing through a fixed point & having a known slope (D) a circle with a known centre & a given radius. Q. All the points lying inside the triangle formed by the points (, ), (5, 6) & (, ) satisfy x + y 0 (B) x + y + 0 (C) x + y 0 (D) x + 0 Q.4 The equations of the tangents drawn from the origin to the circle, x² + y² rx hy + h² = 0 are x = 0 (B) y = 0 (C) (h² r²) x rhy = 0 (D) (h² r²)x + rhy = 0 Q.5 The coordinates of the fourth vertex of the parallelogram where three of its vertices are (, 4); (0, 4) & (5, ) can be : (8, 6) (B) (, 0) (C) ( 8, ) Q.6 The equation of a circle with centre (4, ) and touching the circle x + y = is : x + y 8x 6y 9 = 0 (B) x + y 8x 6y + = 0 (C) x + y 8x 6y = 0 (D) x + y 8x 6y + 9 = 0 Q.7 Two vertices of the Δ ABC are at the points A(, ) and B(4, 5) and the third vertex lines on the straight line y = 5(x ). If the area of the Δ is 9/ then the possible co ordinates of the vertex C are: (5, 0) (B) (, 0) (C) (, 5) (D) (5, 4) Q.8 A circle passes through the points (, ), (0, 6) and (5, 5). The point(s) on this circle, the tangent(s) at which is/are parallel to the straight line joining the origin to its centre is/are : (, 5) (B) (5, ) (C) ( 5, ) (D) (, 5) Q.9 Line x y = cuts the co ordinate axes at A(a, 0) & B (0, b) & the line x y a b a = at b A ( a, 0) & B (0, b ). If the points A, B, A, B are concyclic then the orthocentre of the triangle ABA is: (0, 0) (B) (0, b') (C) 0, aa bb' (D) 0, b a Q.40 Point M moved along the circle (x 4) + (y 8) = 0. Then it broke away from it and moving along a tangent to the circle, cuts the x axis at the point (, 0). The co ordinates of the point on the circle at which the moving point broke away can be : 46, (B) , (C) (6, 4) (D) (, 5) 5 5 Q.4 If one vertex of an equilateral triangle of side 'a' lies at the origin and the other lies on the line x y = 0 then the coordinates of the third vertex are : a a (0, a) (B), a a (C) (0, a) (D), Q.4 The circles x + y + x + 4y 0 = 0 & x + y + 6x 8y + 0 = 0 are such that the number of common tangents on them is (B) are not orthogonal (C) are such that the length of their common tangent is 5 (/5) /4 (D) are such that the length of their common chord is 5. [4]
15 Q.4 Two straight lines u = 0 and v = 0 passes through the origing forming an angle of tan (7/9) with each other. If the ratio of the slopes of u = 0 and v = 0 is 9/ then their equations are: y = x & y = x (B) y = x & y = x (C) y + x = 0 & y + x = 0 (D) y + x = 0 & y + x = 0 Q.44 The centre(s) of the circle(s) passing through the points (0, 0), (, 0) and touching the circle x + y =9 is/are :, (B), (C), / (D) /, Q.45 Given two straight lines x y 7 = 0 and x y + = 0. Equation of a line which divides the distance between them in the ratio : can be : x y = 0 (B) x y = 0 (C) y = x (D) x y + = 0 Q.46 The circles x + y x 4y + = 0 and x + y + 4x + 4y = 0 touch internally (B) touch externally (C) have x + 4y = 0 as the common tangent at the point of contact. (D) have x + 4y + = 0 as the common tangent at the point of contact. Q.47 Three vertices of a triangle are A(4, ) ; B(, ) and C(7, k). Value(s) of k for which centroid, orthocentre, incentre and circumcentre of the Δ ABC lie on the same straight line is/are : 7 (B) (C) 9/8 Q.48 A and B are two fixed points whose coordinates are (, ) and (5, 4) respectively. The coordinates of a point P if ABP is an equilateral triangle, is/are : 4 4 4, 4, (B), (C), (D) Q.49 Which of the following lines have the intercepts of equal lengths on the circle, x + y x + 4y = 0? x y = 0 (B) x + y = 0 (C) x + y + 0 = 0 (D) x y 0 = 0 Q.50 Straight lines x + y = 5 and x y = intersect at the point A. Points B and C are chosen on these two lines such that AB = AC. Then the equation of a line BC passing through the point (, ) is x y = 0 (B) x + y = 0 (C) x + y 9 = 0 (D) x y + 7 = 0 Q.5 Equation of a straight line passing through the point (, ) and inclined at an angle of arc tan with the line y + x = 5 is: y = (B) x = (C) x + 4y 8 = 0 (D) 4x + y 7 = 0 Q.5 The x coordinates of the vertices of a square of unit area are the roots of the equation x x + = 0 and the y coordinates of the vertices are the roots of the equation y y + = 0 then the possible vertices of the square is/are : (, ), (, ), (, ), (, ) (B) (, ), (, ), (, ), (, ) (C) (, ), (, ), (, ), (, ) (D) (, ), (, ), (, ), (, ) Q.5 Consider the equation y y = m (x x ). If m & x are fixed and different lines are drawn for different values of y, then the lines will pass through a fixed point (B) there will be a set of parallel lines (C) all the lines intersect the line x = x (D) all the lines will be parallel to the line y = x. [5]
16 ANSWER KEY Select the correct alternative : (Only one is correct) Q. D Q. B Q. A Q.4 A Q.5 A Q.6 B Q.7 B Q.8 A Q.9 C Q.0 B Q. A Q. D Q. C Q.4 C Q.5 B Q.6 A Q.7 B Q.8 C Q.9 D Q.0 B Q. D Q. B Q. B Q.4 A Q.5 C Q.6 A Q.7 B Q.8 C Q.9 A Q.0 A Q. B Q. B Q. B Q.4 D Q.5 B Q.6 A Q.7 A Q.8 C Q.9 D Q.40 D Q.4 B Q.4 D Q.4 A Q.44 B Q.45 B Q.46 C Q.47 A Q.48 A Q.49 D Q.50 A Q.5 D Q.5 D Q.5 B Q.54 C Q.55 A Q.56 C Q.57 B Q.58 A Q.59 B Q.60 B Q.6 D Q.6 A Q.6 B Q.64 B Q.65 D Q.66 B Q.67 D Q.68 B Q.69 D Q.70 D Q.7 D Q.7 D Q.7 A Q.74 B Q.75 C Q.76 C Q.77 B Q.78 D Q.79 C Q.80 C Q.8 D Q.8 A Q.8 C Q.84 A Q.85 A Q.86 A Q.87 B Q.88 C Q.89 A Q.90 C Q.9 D Q.9 A Q.9 D Q.94 B Q.95 D Q.96 D Q.97 B Q.98 B Q.99 A Q.00 D Q.0 D Q.0 A Q.0 C Q.04 D Q.05 D Q.06 C Q.07 B Q.08 C Q.09 C Q.0 D Q. B Q. A Q. A Q.4 B Q.5 B Q.6 C Q.7 A Q.8 A Q.9 A Q.0 B Q. D Q. B Q. D Q.4 A Q.5 B Q.6 C Q.7 A Q.8 A Select the correct alternatives : (More than one are correct) Q.9 B,D Q.0 A,B,D Q. A,D Q. A,B,C,D Q. A,B,D Q.4 A,C Q.5 A,B,C Q.6 C,D Q.7 A,B Q.8 B,D Q.9 B,C Q.40 B,C Q.4 A,B,C,D Q.4 A,C,D Q.4 A,B,C,D Q.44 C,D Q.45 A,B Q.46 B,C Q.47 B,C Q.48 A,B Q.49 A,B,C,D Q.50 A,B Q.5 B,C Q.5 A,B Q.5 B,C [6]
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