= 0 1 (3 4 ) 1 (4 4) + 1 (4 3) = = + 1 = 0 = 1 = ± 1 ]


 Gertrude Poole
 3 years ago
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1 STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. If the lines x + y + = 0 ; x + y + = 0 and x + y + = 0, where + =, are concurrent then (A) =, = (B) =, = ± (C) =, = ± (D*) = ±, = [Sol. Lines are x + y + = 0 ; x + y + = 0 and x + y + = 0, where + = = 0 ( ) ( ) + ( ) = + + = + = 0 = = ± ] Q. Given the family of lines, a (x + y + 6) + b (x + y + ) = 0. The line of the family situated at the greatest distance from the point P (, ) has equation : (A*) x + y + 8 = 0 (B) 5x + y + 0 = 0 (C) 5x + 8y + 0 = 0 (D) none [Hint : point of intersection is A (, 0). The required line will be one which passes through (, 0) and is perpendicular to the line joining (, 0) and (, ) or taking (, ) as centre and radius equal to PA draw a circle, the required line will be a tangent to the circle at (, 0) ] Q. A variable rectangle PQRS has its sides parallel to fixed directions. Q and S lie respectively on the lines x = a, x = a and P lies on the x axis. Then the locus of R is (A*) a straight line (B) a circle (C) a parabola (D) pair of straight lines [Hint: Mid point of QS = Mid point of PS 0 = h + x x = h hence p ( h, 0) equation of PQ y = mx + c passes through ( h, 0) c = mh y = mx + mh since Q lies on it = ma + mh y now m QR = m = y K a h m = m ( a h ) K a h simplifying h + mk = a + (a + h)m a st. line] Q. A rectangular billiard table has vertices at P(0, 0), Q(0, 7), R(0, 7) and S (0, 0). A small billiard ball starts at M(, ) and moves in a straight line to the top of the table, bounces to the right side of the table, then comes to rest at N(7, ). The ycoordinate of the point where it hits the right side, is (A*).7 (B).8 (C).9 (D) none Use Image method Q.5 B and C are points in the xy plane such that A(, ) ; B (5, 6) and AC = BC. Then (A) 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. [Hint: Locus of C is a circle 8(x + y ) 88x 0y + 5 = 0 (D)] Q.B on Straight line and circle []
2 Q.6 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 : (A*) 5x + y = (B) 5x y = (C) x + 5y = (D) x 5y = 6 [Hint : a = 5 a a = 5 Q.7 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 (A) m(m + n) (B*) m(m + n) (C) m(m + n) (D) m(m + n) [Sol. Area of rectangle BCDE = mn m(m n) Area of ABC = = m mn area of pentagon = mn + m mn = m + mn Ans. ] Q.8 The area enclosed by the graphs of x + y = and x = is (A) (B) (C) 6 (D*) 8 [Hint: figure is a parallelogram ] Area = = 8 Ans. ] Q.9 If P = (, 0) ; Q = (, 0) and R = (, 0) are three given points, then the locus of the points S satisfying the relation, SQ + SR = SP is : (A) a straight line parallel to xaxis (B) a circle passing through the origin (C) a circle with the centre at the origin (D*) a straight line parallel to yaxis. Q.0 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 : (A*) x 8y = 0 (B) x + y = 0 (C) 8x + y = 0 (D) x y = 0 [ Hint : (y x) (y mx) = mx xy (m + ) + y = 0 the equation to the pair of bisectors are : x m y = xy m x 7xy y m (m ) = 7 or 8m = m = ] 8 Q. Two points A(x, y ) and B(x, y ) are chosen on the graph of f (x) = ln x with 0 < x < x. The points C and D trisect line segment AB with AC < CB. Through C a horizontal line is drawn to cut the curve at E(x, y ). If x = and x = 000 then the value of x equals [Sol. (A*) 0 (B) 0 (C) (0) / (D) (0) / Using section formula Q.B on Straight line and circle []
3 000 a = = 0 ln000 ln000 b = b = now line y = b intersects the curve y = ln x b = ln x...() from () and ()...() ln000 = ln x ln (000) / = ln x x = (000) / = 0 Ans. ] Q. If L is the line whose equation is ax + by = c. Let M be the reflection of L through the yaxis, and let N be the reflection of L through the xaxis. Which of the following must be true about M and N for all choices of a, b and c? (A) The xintercepts of M and N are equal. (B) The yintercepts of M and N are equal. (C*) The slopes of M and N are equal. (D) The slopes of M and N are reciprocal. [Sol. Reflecting a graph over the xaxis results in the line M whose equation is ax by = c, while a reflection through the yaxis results in the line N whose equation is ax + by = c. Both clearly have slope equal to a/b (from, say, the slopeintercept form of the equation.) ] Q. 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 (A) d d (B*) (C) d d (D) d d [ Hint : x cos ( + 0º) = d () and x sin = d () dividing cot = d, squaring equation () and putting d the value of cot, x = (d d + ) x = d d Q. 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 (A) (B) / (C) / (D*) /8 [Sol. Let the coordinates of C be (, c) c y c mx m = ; m x = x m m x = c m x (m m )x = c m c = (m m )x + m...() ] Q.B on Straight line and circle []
4 now area of ABC = 0 x 0 m x = [cx m x] = c [(( m m)x m)x m x] = [( m m)x mx mx] = (m m )(x x ) (x > x in (0, ) Hence, f (x) = (x x ); f (x)] max = 8 when x = ] Q.5 Consider a parallelogram whose sides are represented by the lines x + y = 0; x + y 5 = 0; x y = 0 and x y =. The equation of the diagonal not passing through the origin, is (A) x y + 5 = 0 (B) 9x y + 5 = 0 (C) x 9y 5 = 0 (D*) x y 5 = 0 [Sol. Equation of AC (x + y)(x y) = (x y )(x + y 5) or 5(x y) + (x + y) 5 = 0 x y 5 = 0 Ans. equation of BD is (x + y)(x y ) = (x y)(x + y 5) 5(x y) (x + y) = 0 9x 9y = 0 ] Q.6 What is the yintercept of the line that is parallel to y = x, and which bisects the area of a rectangle with corners at (0, 0), (, 0), (, ) and (0, )? (A) (0, 7) (B) (0, 6) (C*) (0, 5) (D) (0, ) [Sol. Area of OADE = Area of ABCD since Area of ADD' = Area of A'DA so area of rectangle AA'OE = Area of rectangle D'DCB BD' = OA h = h h + h =...() m AD = 0 h h = h h =...() 0 from () and () h = = h = 5 5 Equation of line AD is y 0 = x put x = 0, get y = 5 (0, 5) Ans. ] 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 (A*) x + y = (B) x + y = (C) x + y = xy (D) x + y = [ Hint: y = m (x ) Q.B on Straight line and circle []
5 y = (x ) m h = m k = + m locus is x + y = ] Q.7 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 (A*) x + y + xy = 0 (B) x + y xy + = 0 (C) x + y + xy + = 0 (D) x + y xy = 0 [Sol. Let (h, k) be point of intersection then h k a b ah kb b a multiply h k hk( ) a b h k hk x y xy 0 given a b a b ab b a Note that the locus is not physically viable ] Q.8 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 (A*) (x y ) = x + y (B) (x + y ) = (x y ) (C) (x + y ) = x y (D) (x y ) = (x + y ) [Sol. Coordinates of circumcentre l m, l m m l h = l l m...() and m k = l m...() also l + m = square and add () and () and then subtract () and () l m h + k = ( l m ) l m and h k = ( l m ) hence h + k = (h k ) locus is x + y = (x y ) ] = ( l m = l m ) Q.B on Straight line and circle [5]
6 Q.9 Area of the triangle formed by the line x + y = and the angle bisectors of the line pair x y + y = 0 is (A*) / (B) (C) / (D) [Sol. x (y y + ) = 0 x (y ) = 0 (x + y )(x y + ) = 0 Area = = Ans. ] Q.0 If the straight lines, ax + amy + = 0, b x + (m + ) b y + = 0 and cx + (m + )cy + = 0, m 0 are concurrent then a, b, c are in : (A) A.P. only for m = (B) A.P. for all m (C) G.P. for all m (D*) H.P. for all m. [Sol. D = a b c am b(m ) = 0 ; c(m ) a b c 0 b c a(b c) +(bc bc) = 0 ab ac + bc = 0 b(a + c) = ac b = ac a c = 0 a, b, c are in H.P. ] Q. A is a point on either of two lines y + x= at a distance of units from their point of intersection. The coordinates of the foot of perpendicular from A on the bisector of the angle between them are [Sol. (A), (B*) (0, 0) (C), (D) (0, ) Draw figure y x for x > 0 y x for x < 0 ] Q. P is a point inside the triangle ABC. Lines are drawn through P, parallel to the sides of the triangle. The three resulting triangles with the vertex at P have areas, 9 and 9 sq. units. The area of the triangle ABC is [Sol. (A) (B) (C) (D*) 9 q = a ly or p a 7 q a...() and...() r a...() () + () + () gives p q r = a = = ] Q.B on Straight line and circle [6]
7 Q. Let PQR be a right angled isosceles triangle, right angled at P (, ). If the equation of the line QR is x + y =, then the equation representing the pair of lines PQ and PR is (A) x y + 8xy + 0x + 0y + 5 = 0 (B*) x y + 8xy 0x 0y + 5 = 0 (C) x y + 8xy + 0x + 5y + 0 = 0 (D) x y 8xy 0x 5y 0 = 0 Q. An equilateral triangle has each of its sides of length 6 cm. If (x, y ) ; (x, y ) and (x, y ) are its vertices then the value of the determinant, x x x y y y is equal to : (A) 9 (B) (C) 86 (D*) 97 [Hint: x x x y y y = A D = A D = 6 = 97 Ans. ] Q.5 A graph is defined in polar coordinates as r() = cos +. The smallest xcoordinates of any point on the graph is (A*) 6 (B) 8 (C) (D) [Sol. xcoordinate = r cos = (cos + ) cos = cos + Hence smallest is 6 Ans. ] cos = (cos + ) 6 Paragraph for Question Nos. 6 to 8 Consider a general equation of degree, as x 0xy + y + 5x 6y = 0 Q.6 The value of '' for which the line pair represents a pair of straight lines is (A) (B*) (C) / (D) Q.7 For the value of obtained in above question, if L = 0 and L = 0 are the lines denoted by the given line pair then the product of the abscissa and ordinate of their point of intersection is (A) 8 (B) 8 (C*) 5 (D) 5 Q.8 If is the acute angle between L = 0 and L = 0 then lies in the interval (A) (5, 60 ) (B) (0, 5 ) (C) (5, 0 ) (D*) (0, 5 ) [Sol. (i) a = ; h = 5; b = ; g = 5 ; f = 8, c = 5 5 ()( ) + ( 8) ( 5) (6) + 5 = = 0 00 = 00 = Ans. Q.B on Straight line and circle [7]
8 (ii) x 0xy + y + 5x 6y = 0 consider the homogeneous part x 0xy + y x 6xy xy + y or x(x y) y(x y) or (x y)(x y) x 0xy + y + 5x 6y (x 6y + A)(x y + B) solving A = ; B = hence lines are x 6y = 0 and x y + = 0 7 solving intersection point 0, product = 5 Ans. (iii) tan = h ab a b = 5 = 7 (0, 5 ) Ans. ] Paragraph for Question Nos. 9 to Consider a family of lines (a + )x (a + )y (a + ) = 0 where a R Q.9 The locus of the foot of the perpendicular from the origin on each member of this family, is (A) (x ) + (y + ) = 5 (B) (x ) + (y + ) = 5 (C) (x + ) + (y ) = 5 (D*) (x ) + (y ) = 5 Q.0 A member of this family with positive gradient making an angle of with the line x y =, is (A*) 7x y 5 = 0 (B) x y + = 0 (C) x + 7y = 5 (D) 5x y = 0 Q. Minimum area of the triangle which a member of this family with negative gradient can make with the positive semi axes, is (A) 8 (B) 6 (C*) (D) [Sol. Given (a + )x (a + )y (a + ) = 0 (x y ) + a(x y ) = 0 family of lines passes through the fixed point P which is the intersection of x y = and x y = solving P(, ), now (i) k k = h h locus is x(x ) + y(y ) = 0 x + y y x = 0 5 x (y ) (x ) + (y ) = 5 Ans. (ii) We have y = m(x )...() this makes an angle of / with x y = with slope / m ( ) m = ; m = ± m = + m (with + ve sign) m = 7 m with ve sign m = m 7m = m = 7 (rejected) hence the line is y = 7(x ) 7x y 5 = 0 Ans. Q.B on Straight line and circle [8]
9 (iii) Again y = m(x ) x = 0; y = m; y = 0, x = m A = ( m) (m < 0) m A = m + = + m m let m = M (M > 0) m A = + M + = M = M M area is minimum if M = m = A min = 8 A min = Ans. ] 8 M M [MULTIPLE OBJECTIVE TYPE] Q. If x y = is a line through the intersection of x y = and x y = and the lengths of the c d a b b a perpendiculars drawn from the origin to these lines are equal in lengths then : (A*) = a b c d (B) (C*) a b = (D) none c d [Hint : Use the condition of concurrency for three lines ] = a b c d Q. A and B are two fixed points whose coordinates are (, ) and (5, ) respectively. The coordinates of a point P if ABP is an equilateral triangle, is/are : (A*), (B*), (C), (D), [ Hint : use parametric ] Q. 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 (A*) x y = 0 (B*) x + y = 0 (C) x + y 9 = 0 (D) x y + 7 = 0 [Hint: Note that the lines are perpendicular. Find the equation of the lines through (, ) and parallel to the bisectors of the given lines, the slopes of the bisectors being / and ] Q.5 The straight lines x + y = 0, x + y = 0 and x + y = 0 form a triangle which is (A*) isosceles (B) right angled (C*) obtuse angled (D) equilateral Q.B on Straight line and circle [9]
10 CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y =. The radius of the circle is [Sol. (A*) 5 (B) 5 (C) 5 (D) 5 x y + = 0 is tangent slope of line OA = equation of OA, (y 5) = (x ) y 0 = x + x + y = intersection with x y = will give coordinates of centre solving we get (8, ) distance OA = ( 8 ) ( 5) = 6 9 = 5 = 5 ] Q. If the circles x + y + ax + cy + a = 0 and x + y ax + dy = 0 intersect in two distinct points P and Q then the line 5x + by a = 0 passes through P and Q for (A) exactly one value of a (B*) no value of a (C) infinitely many values of a (D) exactly two values of a [Sol. S S = 5ax + (c d)y + a + = 0 and 5x + by a = 0 must represent the same line a c d a b a ab = c d and + a + a + = 0 a is imaginary only so no value of a (B) ] Q. Four unit circles pass through the origin and have their centres on the coordinate axes. The area of the quadrilateral whose vertices are the points of intersection (in pairs) of the circles, is [Sol. (A) sq. unit (B) sq. units (C*) sq. units (D) can not be uniquely determined, insufficient data Ar. (square of sides ) = sq. units ] Q.B on Straight line and circle [0]
11 Q. To which of the following circles, the line y x + = 0 is normal at the point,? (A) x y 9 (B) x y 9 (C) x + (y ) = 9 (D*) (x ) + y = 9 [Hint: Line must pass through the centre of the circle ] Q.5 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 [Sol. (A) 0, (B) 0, (C*) 0, (D) none Equation of circle are x + y = ; x + y = ( d) ; x + y = ( d) solve any of circle with line y = x + e.g. x + y = ( d) x + x + d d = 0 cuts the circle in real and distinct point hence > 0 d d + > 0 d = ] Q.6 The circle with equation x + y = intersects the line y = 7x + 5 at two distinct points A and B. Let C be the point at which the positive xaxis intersects the circle. The angle ACB is (A) tan (B) tan (C*) tan () (D) tan [Sol. Solving y = 7x + 5 and the circle x + y = [th, ] A, 5 5 and B m OA =, 5 ; m OB = 5 Hence AOD = 90 ACB = = tan () Ans. ] Q.7 The number of common tangents of the circles (x + )² + (y )² = 9 and (x )² + (y + )² = is (A) 0 (B*) (C) (D) Q.8 In the xyplane, the length of the shortest path from (0, 0) to (, 6) that does not go inside the circle (x 6) + (y 8) = 5 is 5 (A) 0 (B) 0 5 (C*) 0 + [Sol. Let O = (0, 0), P = (6, 8) and Q = (, 6). As shown in the figure the shortest route consists of tangent OT, minor arc TR and tangent RQ. Since OP = 0, PT = 5, and OTP = 90, (D) Q.B on Straight line and circle []
12 it follows that OPT = 60 and OT = 5. By similar reasoning, QPR = 60 and QR = 5. Because O, P and Q are collinear (why?), 5 RPT = 60, so arc TR is of length. 5 Hence the length of the shortest route is ( 5 ) + Ans. ] Q.9 Triangle ABC is right angled at A. The circle with centre A and radius AB cuts BC and AC internally at D and E respectively. If BD = 0 and DC = 6 then the length AC equals (A) 6 (B*) 6 6 (C) 0 (D) [Sol. b + r = (6)...() Also CD CB = CE CX (using power of the point C) 6 6 = (b r)(b + r) b r = 6 6 from () and () b = 6(6 + 6) = 6 5 b = 6 6 b = 6 6 Ans. ] Q.0 The equation of a line inclined at an angle to the axis X, such that the two circles x + y =, x + y 0x y + 65 = 0 intercept equal lengths on it, is (A*) x y = 0 (B) x y + = 0 (C) x y + 6 = 0 (D) x y 6 = 0 [Sol. Let equation of line be y = x + c y x = c...() perpendicular from (0, 0) on () is c = c In AON, c = AN and in CPM, c = CM perpendicular from (5, 7) on line y x = c = c Given AN = CM = c = 9 ( c) c = equation of line y = x of x y = 0 ] Q. If x = is the chord of contact of the circle x y = 8, then the equation of the corresponding pair of tangents, is (A) x 8y + 5x + 79 = 0 (B*) x 8y 5x + 79 = 0 (C) x 8y 5x 79 = 0 (D) x 8y = 79 Q.B on Straight line and circle []
13 [Sol. Equation of tangent to the circle at A,6 x + and at B, 6 6 y = 8...() x 6 y = 8...() hence equation of the line pair (x + y 7)(x y 7) = 0 [th, 007] x 8y 7(x + y ) 7(x y ) + 79 = 0 x 8y 5x + 79 = 0 Ans.] Q. Let C and C are circles defined by x + y 0x + 6 = 0 and x + y + 0x + = 0. The length of the shortest line segment PQ that is tangent to C at P and to C at Q is (A) 5 (B) 8 (C*) 0 (D) [Sol. Centres are (0, 0) and ( 5, 0) r = 6; r = 9 d = 5 r + r < d circles are separated PQ = l = d (r r ) = 65 5 = 0 Ans. ] Q. If the straight line y = mx is outside the circle x + y 0y + 90 = 0, then (A) m > (B) m < (C) m > (D*) m < [Sol. Centre (0,0), radius 0. 0 Distance of (0,0) from y = mx is greater than 0 i.e. 0 < m < ] m Q. The centre of the smallest circle touching the circles x + y y = 0 and x + y 8x 8y + 9 = 0 is (A) (, ) (B) (, ) (C) (, 7) (D*) (, 5) Q.5 Suppose that two circles C and C in a plane have no points in common. Then (A) there is no line tangent to both C and C. (B) there are exactly four lines tangent to both C and C. (C) there are no lines tangent to both C and C or there are exactly two lines tangent to both C and C. (D*) there are no lines tangent to both C and C or there are exactly four lines tangent to both C and C. [Hint: or ] Q.6 A variable line moves in such way that the product of the perpendiculars from (a, 0) and (0, 0) is equal to k. The locus of the feet of the perpendicular from (0, 0) upon the variable line is a circle, the square of whose radius is (Given: a < k ) (A*) a a k k (B) (C) a k + (D) k Q.B on Straight line and circle [] a
14 Sol. Let the equation of the variable line is [th, , P] xx + yy (x + y ) = 0 x y p p = x y ax (x x y i.e. x + y ax = k locus is x + y ax ± k = 0 y ) = k (x, y ) Variable line x m= y r a = (± k ) +ve sign r = p p (0,0) (a,0) a k (not possible as r becomes ve) ve sign r a = + k Ans. ] Q.7 The shortest distance from the line x + y = 5 to the circle x + y = 6x 8y is equal to (A*) 7/5 (B) 9/5 (C) /5 (D) /5 [Sol. Centre: (, ) and r = 5 perpendicular distance from (, ) on x + y 5 = 0 is p = = 5 7 d = 5 = Ans. ] 5 5 Q.8 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 /, then the coordinates of the centre of C are 9 (A), (B*) , 5 (C) 9, (D) 5 5 [Hint : Equation of common chord is y = x x y = 0 family of circle x + y + (x y) = 0 equate radius of this circle as 5. Alternately...] Q.9 The points (x, y ), (x, y ), (x, y ) and (x, y ) are always : (A) collinear (B*) concyclic (C) vertices of a square (D) vertices of a rhombus [Hint: All the points lie on the circle (x x ) (x x ) + (y y ) (y y ) = 0 Note that points form a rectangle] 5 9, 5 Q.0 Let C be the circle of radius unity centred at the origin. If two positive numbers x and x are such that the line passing through (x, ) and (x, ) is tangent to C then (A*) x x = (B) x x = (C) x + x = (D) x x = Q.B on Straight line and circle []
15 [Sol. Let the equation tangent in parametric form is x cos + y sin = put (x, ) x cos sin = x cos = + sin...() now put (x, ) x cos = sin...() () () x x cos = sin = cos x x = Ans. Q. 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 (A) r (B*) r (C) r (D) r 5 [HInt: ] Q. The locus of the middle points of the system of chords of the circle x² + y² = 6 which are parallel to the line y = x + 5 is (A) x = y (B*) x + y = 0 (C) y + x = 0 () y = x Q. The distance between the chords of contact of tangents to the circle x + y + gx + fy + c = 0 from the origin and the point (g, f) is [Sol. (A) g f (B) g f c (C*) g f c g f Equation of chord or contact of the pair of tangent from (0, 0) and (g, f) are gx + fy + c = 0...() and gx + fy + g f (D) c = 0...() g f c g f These lines are parallel hence distance = g c g f f c ] Q. The locus of the center of the circles such that the point (, ) is the mid point of the chord 5x + y = 6 is (A*) x 5y + = 0 (B) x + 5y = 0 (C) x + 5y + = 0 (D) none [ Hint : Slope of the given line = 5/ 5 f. g = 5 + 5f = + g locus is x 5y + = 0 ] Q.B on Straight line and circle [5]
16 Q.5 The points A (a, 0), B (0, b), C (c, 0) and D (0, d) are such that ac = bd and a, b, c, d are all nonzero. Then the points (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D*) are concyclic [Sol. If OA OC = OB OD (Power of point) then points are concylic a c = bd (true) ] Q.6 The locus of the centers of the circles which cut the circles x + y + x 6y + 9 = 0 and x + y 5x + y = 0 orthogonally is : (A) 9x + 0y 7 = 0 (B) x y + = 0 (C*) 9x 0y + = 0 (D) 9x + 0y + 7 = 0 [Hint: Let out circle be x + y + gx + fy + c = 0 conditions ( g) ( ) + ( f ) () = c + 9 and ( g) (5/) + ( f ) ( ) = c ag 0 f = locus of centre 9x 0y + = 0 ] Q.7 If the angle between the tangents drawn from P to the circle x + y + x 6y + 9 sin + cos =0 is, then the locus of P is (A) x + y + x 6y + = 0 (B) x + y + x 6y 9 = 0 (C) x + y + x 6y = 0 (D*) x + y + x 6y + 9 = 0 [Sol. C(, ); R = sin cos = sin R = sin (,)C sin Now use sin = CP = CP Hence locus is (x + ) + (y ) = ] P(x,y) y sin Q.8 The locus of the mid points of the chords of the circle x² + y² + x 6y = 0 which subtend an angle of radians at its circumference is : (A) (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 [ Hint : radius = 5 p = 5 cos 60º =.5 locus is (h + ) + (k ) = 6.5 ] Q.9 The angle at which the circles (x ) + y = 0 and x + (y ) = 5 intersect is (A) (B*) 6 (C) (D) [Hint: cos = = = ] Q.B on Straight line and circle [6]
17 Q.0 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 (A) (, 0) (B) (0, ) (C) (0, ) (D*) (, 0) [Sol. The particle which moves clockwise is moving three times as fast as the particle moving anticlockwise This tells us that in the time that the clockwise particle travels (/) th of the way around the circle the anticlockwise particle will travel (/) th of the way around the circle and so the nd particle will meet at p (0, ). Using the same logic they will meet at Q (, 0) when they meet the nd time] Q. 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 : (A) (, ] [, ) (B) {, } (C) { } (D*) { } [Hint: x + y + x = 0; centre (, 0) and rad. = line x y + c = 0 c ; c = ; c = ± c = or ] Q. Locus of the middle points of a system of parallel chords with slope, of the circle x + y x y = 0, has the equation (A*) x + y = 0 (B) x y = 0 (C) x y = 0 (D) x + y 5 = 0 [Hint: Locus will be a line with slope / and passing through the centre (, ) of the circle y = (x ) y = x + x + y = 0 Ans. ] Q. Two circles are drawn through the points (, 0) and (, ) to touch the axis of y. They intersect at an angle (A*) cot (B) cos 5 (C) (D) tan [Hint : The two circles are x + y x + y + = 0 and x + y 0x 6y + 9 = 0 (use family of circles) ] Q. A foot of the normal from the point (, ) to a circle is (, ) and a diameter of the circle has the equation x y = 0. Then the equation of the circle is (A) x + y y + = 0 (B) x + y y + = 0 (C*) x + y x = 0 (D) x + y x + = 0 [Sol. Diameter of the circle is given as x y = 0...() slope of PN: = Q.B on Straight line and circle [7]
18 equation of normal through PN is y = (x ) x y = 0...() solving () and () centre is (, 0) hence equation of the circle is (x ) + y = ( ) + x + y x = 0 Ans. ] Q.5 A rhombus is inscribed in the region common to the two circles x + y x = 0 and x + y + x = 0 with two of its vertices on the line joining the centres of the circles. The area of the rhombous is : (A*) 8 sq.units (B) sq.units (C) 6 sq.units (D) none [Hint : circles with centre (, 0) and (, 0) each with radius y axis is their common chord. The inscribed rhombus has its diagonals equal to and A = d d = 8 ] Q.6 A circle is drawn touching the xaxis and centre at the point which is the reflection of (a, b) in the line y x = 0. The equation of the circle is (A) x + y bx ay + a = 0 (B*) x + y bx ay + b = 0 (C) x + y ax by + b = 0 (D) x + y ax by + a = 0 k b [Sol. h a also a h and b k a b = k h and touches the xaxis k b = h a or a b = h k lies on yaxis g c = 0 a h = b k centre (b, a) g = c = b equation x + y hx ay + b = 0 ] Q.7 A(, 0) and B(0, ) and two fixed points on the circle x + y =. C is a variable point on this circle. As C moves, the locus of the orthocentre of the triangle ABC is (A*) x + y x y + = 0 (B) x + y x y = 0 (C) x + y = (D) x + y + x y + = 0 [Sol. Let C (cos, sin ); H(h, k) is the orthocentre of the ABC B(0, ) h = + cos k = + sin (x ) + (y ) = x + y x y + = 0 ] C(cos, sin ) O A(, 0) Q.B on Straight line and circle [8]
19 Q.8 A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If d and 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 : (A) d d (B) [Sol. Let the circle be x + y + gx + fy = 0 Tangent at the origin is gx + fy = 0 d = d = g g g f f f d d (C*) d + d (D) d d g f diameter d d d d Q.9 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 (A) AB AD AB AD (B) [Sol. l x = y l x l x = y (l + x ) l (x y ) = x y l = x xy y = AB AD AB AD (C) where l = AC; x = AB, y = AD AB AD AB AD Ans. ] AB AD (D*) AB AD AB AD Q.0 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 (A) (B) 5 (C*) 0 (D) 5 [Sol. slope of l = slope of l = equation of l y = (x 0) y + x = 0 Hence t = 0 Ans.] Q. B and C are fixed points having coordinates (, 0) and (, 0) respectively. If the vertical angle BAC is 90º, then the locus of the centroid of the ABC has the equation : (A*) x + y = (B) x + y = (C) 9 (x + y ) = (D) 9 (x + y ) = [Hint : Let A (a, b) and G (h. k) Now A, G, O are collinear h = 0. a a = h and similarly b = k. Now (a, b) lies on the circle x + y = 9 A ] ] Q.B on Straight line and circle [9]
20 Paragraph for Question Nos. to Let C be a circle of radius r with centre at O. Let P be a point outside C and D be a point on C. A line through P intersects C at Q and R, S is the midpoint of QR. Q. For different choices of line through P, the curve on which S lies, is (A) a straight line (B) an arc of circle with P as centre (C) an arc of circle with PS as diameter (D*) an arc of circle with OP as diameter Q. Let P is situated at a distance 'd' from centre O, then which of the following does not equal the product (PQ) (PR)? (A) d r (B) PT, where T is a point on C and PT is tangent to C (C) (PS) (QS)(RS) (D*) (PS) Q. Let XYZ be an equilateral triangle inscribed in C. If,, denote the distances of D from vertices X, Y, Z respectively, the value of product ( + ) ( + ) ( + ), is [Sol. (i) (A*) 0 (B) 8 (C) 6 Locus of S is a part of circle with OP as diameter passing inside the circle 'C' C (D) None of these N Q O M R S(h,k) (ii) (PR) (PQ) = PT = (PN)(PM) = (d r) (d + r) = d r = (PS SR) (PS + SQ) = PS SQ ( SQ = SR) = PS (SQ)(SR) (PQ) (PR) (PS) (iii) Using Ptolemy's theorem, (YD) (XZ) = (XY)(ZD) + (YZ) (XD) = XZ (ZD + XD) { (XY = YZ = ZX) } = + (A) Alternatively: P Y X D Z (YD) (XY) = (YD) from XYD (YD) = + (YD) (XY)...() (YD) = + (YD) (YZ)...() ()() ( ) = = + ] (YD) (YZ) = (YD) from YZD Q.B on Straight line and circle [0]
21 Paragraph for question nos. 5 to 7 Consider circles S : x + y + x = 0 S : x + y = 0 S : x + y + y = 0 Q.5 The radius of the circle which bisect the circumferences of the circles S = 0 ; S = 0 ; S = 0 is (A) (B) (C*) (D) 0 Q.6 If the circle S = 0 is orthogonal to S = 0 ; S = 0 and S = 0 and has its centre at (a, b) and radius equals to 'r' then the value of (a + b + r) equals (A) 0 (B) (C) (D*) Q.7 The radius of the circle touching S = 0 and S = 0 at (, 0) and passing through (, ) is (A) (B) (C*) (D) S S [Sol. (,0) O (,0) (i) S S = 0 (,0) r (a,b) r (0,0) (0, ) S r = a + b + = (a + ) + b + and (a + ) + b + = a + (b + ) + a + = 0 a = b a = b = r = 9 r = Ans. (ii) S S = 0 x = S S y = Radical centre = (, ) radius L T = S = equation of circle is (x ) + (y ) = radius = and a = ; b = a + b + r = Ans. Q.B on Straight line and circle []
22 x = (,) (iii) (,0) family of circles touches the line x = 0 at (, 0) is (x ) + (y 0) + (x ) = 0 passing through (, ) + + = 0 = x + y 6x + 5 = 0 radius 9 5 = Ans. ] Paragraph for Question Nos. 8 to 50 Consider a line pair ax + xy y 5x + 5y + c = 0 representing perpendicular lines intersecting each other at C and forming a triangle ABC with the xaxis. Q.8 If x and x are intercepts on the xaxis and y and y are the intercepts on the yaxis then the sum (x + x + y + y ) is equal to (A) 6 (B*) 5 (C) (D) Q.9 Distance between the orthocentre and circumcentre of the triangle ABC is (A) (B) (C*) 7/ (D) 9/ Q.50 If the circle x + y y + k = 0 is orthogonal with the circumcircle of the triangle ABC then 'k' equals (A) / (B) (C) (D*) / [Sol. (i) As lines are perpendicular a = 0 a = (coefficient of x + coefficient of y = 0) using = 0 c = (D abc + fgh af bg ch ) hence the two lines are x + y = 0 and x y + = 0 x intercepts y intercepts (ii) (CM) = (iii) = x ; x y ; y = = = = Circumcircle of ABC 9 5 x + x + y + y = 5 Ans. CM = 7 Ans. x (x ) y 0 (x + )(x ) + y = 0 (x + y ) 5x = 0 x + y 5 x = 0...() given x + y y + k = 0 which is orthogonal to () using the condition of orthogonality we get, = k k = Ans. ] Q.B on Straight line and circle []
23 Paragraph for question nos. 5 to 5 Consider a circle x + y = and a point P(, ). denotes the angle enclosed by the tangents from P on the circle and A, B are the points of contact of the tangents from P on the circle. Q.5 The value of lies in the interval (A) (0, 5 ) (B) (5, 0 ) (C) 0, 5 ) (D*) (5, 60 ) Q.5 The intercept made by a tangent on the xaxis is (A) 9/ (B*) 0/ (C) / (D) / Q.5 Locus of the middle points of the portion of the tangent to the circle terminated by the coordinate axes is (A*) x + y = (B) x + y = (C) x + y = (D) x y = [Sol. Tangent y = m(x ) mx y + ( m) = 0 m p = m = ( m) = + m m m = 0 m = 0 or m = tan = (5, 60 ) Hence equation of tangent is y = and (with infinite intercept on xaxis) or y = (x ) y 6 = x 6 x y 0 = 0 0 xintercept = Ans.(ii) (B) Variable line with mid point (h, k) x h y, it touches the circle x k + y = h k = h locus is x k + y = Ans.(iii) (A)] [MULTIPLE OBJECTIVE TYPE] Q.5 Let L be a line passing through the origin and L be the line x + y =. If the intercepts made by the circle x + y x + y = 0 on L and L are equal then the equation of L can be (A) x + y = 0 (B*) x y = 0 (C*) x + 7y = 0 (D) x 7y = 0 [Sol. The chords are of equal length, then the distances of the centre from the lines are equal. Let L be y mx = 0. Centre is, m m 7m 6m = 0 m =, ] 7 Q.B on Straight line and circle []
24 Q.55 Consider the circles S : x + y = and S : x + y x y + = 0 which of the following statements are correct? (A*) Number of common tangents to these circles is. (B*) If the power of a variable point P w.r.t. these two circles is same then P moves on the line x + y = 0. (C) Sum of the yintercepts of both the circles is 6. (D*) The circles S and S are orthogonal. [Sol. S : x + y = and S = x + y x y + = 0 centre: (0, 0); radius = centre : (, ); radius = (A) d = distance between centres = 5 r + r = r r = r r < d < r + r these circles are intersecting. number of common tangents is. (A) is correct (B) P(h, k) power of point P is same w.r.t. these two circles. h k = h k h k = h k + h + k 8 = 0 x + y = 0 (B) is correct (C) y intercept of S is = y intercept of S is = 0 sum of yintercept = (C) is incorrect (D) (0 + 0) = + circle is orthogonal. (D) is correct] Q.56 If al bm + dl + = 0, where a, b, d are fixed real numbers such that a + b = d then the line lx + my + = 0 touches a fixed circle : (A*) which cuts the xaxis orthogonally (B) with radius equal to b (C*) on which the length of the tangent from the origin is d b (D) none of these. [Hint : (d b) l + dl + = bm d l + dl + = b (l + m ) d m = b centre (d, 0) and radius b (x d) + y = b ] Q.57 Three distinct lines are drawn in a plane. Suppose there exist exactly n circles in the plane tangent to all the three lines, then the possible values of n is/are (A*) 0 (B) (C*) (D*) [Sol. Case: If lines form a triangle then n = i.e. excircles and incircle Case: If lines are concurrent or all parallel then n = 0 Case: If two are parallel and third cuts then n = hence (A), (C), (D) ] Q.B on Straight line and circle []
25 Q.58 Consider the circles S : x + y + x + y + = 0 S : x + y x + = 0 S : x + y + 6y + 5 = 0 Which of this following statements are correct? (A) Radical centre of S, S and S lies in st quadrant. (B*) Radical centre of S, S and S lies in th quadrants. (C*) Radius of the circle intersecting S, S and S orthogonally is. (D*) Circle orthogonal to S, S and S has its x and y intercept equal to zero. [Hint: radical centre is (, ) radius of the circle orthogonally to S, S, S is and has the equation x + y x + y + = 0 ] Q.59 Locus of the intersection of the two straight lines passing through (, 0) and (, 0) respectively and including an angle of 5 can be a circle with [Sol. m = (A) centre (, 0) and radius. (B) centre (, 0) and radius. (C*) centre (0, ) and radius. (D*) centre (0, ) and radius. k k k h ; m = k h ; tan 5 = h h k h h + k = ± k locus is x + y ± y = 0 k(h h ) = h k x + (y ± ) = (0, ) or (0, ) and radius = (C) and (D) Ans. ] [MATCH THE COLUMN] Q.60 ColumnI ColumnII (A) If the straight line y = kx K I touches or passes outside (P) the circle x + y 0y + 90 = 0 then k can have the value (B) Two circles x + y + px + py 7 = 0 (Q) and x + y 0x + py + = 0 intersect each other orthogonally then the value of p is (C) If the equation x + y + x + = 0 and x + y y + 8 = 0 (R) represent real circles then the value of can be (D) Each side of a square is of length. The centre of the square is (, 7). (S) 5 One diagonal of the square is parallel to y = x. The possible abscissae of the vertices of the square can be [Ans. (A) P, Q, R; (B) Q, R; (C) Q, R, S; (D) P, S] [Sol. (A) x + k x 0kx + 90 = 0 x ( + k ) 0kx + 90 = 0 Q.B on Straight line and circle [5]
26 D 0 00k 90( + k ) 0 0k 9 9k 0 k 9 0 k [, ] p p (B) 5 p = 6 5p + p + 6 = 0 p 5p + 6 = 0 p = or Ans. (C) r = 0 (, ] [, )...() r = (, ] [, )...() () () is (, ] [, ) Ans. (D) Ans. {, 5}] Q.6 ColumnI ColumnII (A) Two intersecting circles (P) have a common tangent (B) Two circles touching each other (Q) have a common normal (C) Two non concentric circles, one strictly inside (R) do not have a common normal the other (D) Two concentric circles of different radii (S) do not have a radical axis. [Ans. (A) P, Q; (B) P, Q; (C) Q; (D) Q, S] Q.6 ColumnI ColumnII (A) Let 'P' be a point inside the triangle ABC and is equidistant (P) centroid from its sides. DEF is a triangle obtained by the intersection of the external angle bisectors of the angles of the ABC. With respect to the triangle DEF point P is its (B) Let 'Q' be a point inside the triangle ABC (Q) orthocentre If (AQ)sin A = (BQ)sin B = (CQ)sin C then with respect to the triangle ABC, Q is its (C) Let 'S' be a point in the plane of the triangle ABC. If the point is (R) incentre such that infinite normals can be drawn from it on the circle passing through A, B and C then with respect to the triangle ABC, S is its (D) Let ABC be a triangle. D is some point on the side BC such that (S) circumcentre the line segments parallel to BC with their extremities on AB and AC get bisected by AD. Point E and F are similarly obtained on CA and AB. If segments AD, BE and CF are concurrent at a point R then with respect to the triangle ABC, R is its [Ans. (A) Q; (B) R; (C) S; (D) P] Q.B on Straight line and circle [6]
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