# (D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2

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1 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 = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5 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 (A) sq. unit (B) sq. units (C) 4 sq. units (D) can not be uniquely determined, insufficient data Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point 3 3 3,? (A) x 3 y 9 (B) x y 9 (C) x + (y 3) = 9 (D) (x 3) + y = 9 Q.4 Let C be a circle x + y =. The line l intersects C at the point (, 0) and the point P. Suppose that the slope of the line l is a rational number m. Number of choices for m for which both the coordinates of P are rational, is (A) 3 (B) 4 (C) 5 (D) infinitely many 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 (A) 0, (B) 4 0, (C) 0, 4 (D) none Q.6 In the xy-plane, 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 3 (B) 0 5 (C) (D) Q.7 If a circle of constant radius 3k passes through the origin 'O' and meets co-ordinate axes at A and B then the locus of the centroid of the triangle OAB is (A) x + y = (k) (B) x + y = (3k) (C) x + y = (4k) (D) x + y = (6k) Q.8 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) 30 (D) 3 Q.9 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) 4 4 (C) a k + 4 (D) a k

2 Q.0 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 Q. 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) 4x x = 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 3 (D) 4r 3 5 Q.3 The locus of the middle points of the system of chords of the circle x² + y² = 6 which are parallel to the line y = 4x + 5 is (A) x = y (B) x + y = 0 (C) y + x = 0 (4) y = x Q.4 The locus of the center of the circles such that the point (, 3) 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 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 non-zero. Then the points (A) form a parallelogram (B) do not lie on a circle (C) form a trapezium (D) are concyclic Q.6 If the angle between the tangents drawn from P to the circle x + y + 4x 6y + 9 sin + 3 cos =0 is, then the locus of P is (A) x + y + 4x 6y + 4 = 0 (B) x + y + 4x 6y 9 = 0 (C) x + y + 4x 6y 4 = 0 (D) x + y + 4x 6y + 9 = 0 Q.7 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 : 3 (A) (x )² + (y + 3)² = 6.5 (B) (x + )² + (y 3)² = 6.5 (C) (x + )² + (y 3)² = 8.75 (D) (x + )² + (y + 3)² = 8.75 Q.8 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 3v. 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) Q.9 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) (, ] [3, ) (B) {, 3} (C) { 3} (D) { } 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 y-axis at P (0, t). The value of 't' is (A) (B) 5 (C) 0 (D) 5

3 Q. The locus of the centre of circle which touches externally the circle x + y 6x 6y + 4 = 0 and also touches the y -axis is (A) x 6x 0y + 4 = 0 (B) x 0x 6y + 4 = 0 (C) y 6x 0y + 4 = 0 (D) y 0x 6y + 4 = 0 Q. A circle of constant radius ' a ' passes through origin ' O ' and cuts the axes of coordinates in points P and Q, then the equation of the locus of the foot of perpendicular from O to PQ is : (A) (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.3 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 Q.4 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 = 4 (D) x + y + x y + = 0 Q.5 A line meets the co-ordinate 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) d d (C) d + d (D) Q.6 AB is a diameter of a circle and C is any point on the circumference of the circle. Then (A) Area of ABC is maximum when it is isosceles. (B) Area of ABC is minimum when it is isosceles. (C) Perimeter of ABC is minimum when it is isosceles. (D) None A d d d d Q.7 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) AB AD AB AD (C) AB AD (D) C D AB AD AB AD Q.8 A variable circle C has the equation x + y (t 3t + )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 (A) 0, 0 (B), 0 0 (C) 0, 0 (D), 0 Q.9 B and C are fixed points having coordinates (3, 0) and ( 3, 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 ) = 4 0

4 5 Q.30 The number of tangents that can be drawn from the point, to the circle passing through the points, 3,, 3 and, 3 3 is (A) (B) 0 (C) (D) None Q.3 If the circles x + y + ax + cy + a = 0 and x + y 3ax + 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 Q.3 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 x-axis intersects the circle. The angle ACB is 4 (A) tan 3 3 (B) tan 4 3 (C) tan () (D) tan Q.33 The number of common tangents of the circles (x + )² + (y )² = 49 and (x )² + (y + )² = 4 is (A) 0 (B) (C) (D) 3 Q.34 The equation of a line inclined at an angle to the axis X, such that the two circles 4 x + y = 4, x + y 0x 4y + 65 = 0 intercept equal lengths on it, is (A) x y 3 = 0 (B) x y + 3 = 0 (C) x y + 6 = 0 (D) x y 6 = 0 Q.35 If x = 3 is the chord of contact of the circle x y = 8, then the equation of the corresponding pair of tangents, is (A) x 8y + 54x + 79 = 0 (B) x 8y 54x + 79 = 0 (C) x 8y 54x 79 = 0 (D) x 8y = 79 Q.36 Let C and C are circles defined by x + y 0x + 64 = 0 and x + y + 30x + 44 = 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) 4 Q.37 If the straight line y = mx is outside the circle x + y 0y + 90 = 0, then (A) m > 3 (B) m < 3 (C) m > 3 (D) m < 3 Q.38 The centre of the smallest circle touching the circles x + y y 3 = 0 and x + y 8x 8y + 93 = 0 is (A) (3, ) (B) (4, 4) (C) (, 7) (D) (, 5) Q.39 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. Q.40 The shortest distance from the line 3x + 4y = 5 to the circle x + y = 6x 8y is equal to (A) 7/5 (B) 9/5 (C) /5 (D) 3/5

5 Q.4 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 3/4, then the co-ordinates of the centre of C are 9 (A), (B) , 5 (C) 9, (D) , 5 Q.4 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 (A) g f (B) g f c (C) g f c g f (D) g f c g f Q.43 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 : (A) 9x + 0y 7 = 0 (B) x y + = 0 (C) 9x 0y + = 0 (D) 9x + 0y + 7 = 0 Q.44 If two distinct chords, drawn from the point (p, q) on the circle x + y = px + qy, where pq 0, are bisected by the x-axis, then (A) p = q (B) p = 8q (C) p < 9q (D) p > 8q Q.45 The angle at which the circles (x ) + y = 0 and x + (y ) = 5 intersect is (A) (B) (C) (D) Q.46 Locus of the middle points of a system of parallel chords with slope, of the circle x + y 4x y 4 = 0, has the equation (A) x + y 4 = 0 (B) x y = 0 (C) x y 3 = 0 (D) x + y 5 = 0 Q.47 If a,, b,, c, and d, are four distinct points on a circle of radius 4 units then, a b c d abcd is equal to (A) 4 (B) /4 (C) (D) 6 Q.48 The radical centre of three circles taken in pairs described on the sides of a triangle ABC as diametres is the : (A) centroid of the ABC (B) incentre of the ABC (C) circumcentre o the ABC (D) orthocentre of the ABC Q.49 Two circles are drawn through the points (, 0) and (, ) to touch the axis of y. They intersect at an angle (A) cot 3 4 (B) cos 4 5 (C) (D) tan Q.50 A foot of the normal from the point (4, 3) 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 4y + = 0 (B) x + y 4y + = 0 (C) x + y x = 0 (D) x + y x + = 0 Q.5 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 : (A) 8 3 sq.units (B) 4 3 sq.units (C) 6 3 sq.units (D) none

6 [REASONING TYPE] Q.5 Consider the circle C : x + y x y 3 = 0 and a point P(3, 4). Statement-: No normal can be drawn to the circle C, passing through (3, 4). Statement-: Point P lies inside the given circle, C. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.53 Passing through a point A(6, 8) a variable secant line L is drawn to the circle S : x + y 6x 8y + 5 = 0. From the point of intersection of L with S, a pair of tangent lines are drawn which intersect at P. Statement-: Locus of the point P has the equation 3x + 4y 40 = 0. Statement-: Point A lies outside the circle. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.54 Consider the lines L : (k + 7)x (k )y 4(k 5) = 0 where k is a parameter and the circle C : x + y + 4x + y 60 = 0 Statement-: Every member of L intersects the circle 'C' at an angle of 90 Statement-: Every member of L is tangent to the circle C. (A) Statement- is true, statement- is true; statement- is correct explanation for statement-. (B) Statement- is true, statement- is true; statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.55 Consider the circles, S : x + y + x 4 = 0 and S : x + y y + = 0 Statement-: Tangents from the point P(0, 5) on S and S are equal. Statement-: Point P(0, 5) lies on the radical axis of the two circles. (A) Statement- is true, statement- is true; statement- is correct explanation for statement-. (B) Statement- is true, statement- is true; statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.56 Statement-: The circle C : x + y 6x 4y + 9 = 0 bisects the circumference of the circle C : x + y 8x 6y + 3 = 0. Statement-: Centre of the circle C lies on the circumference of C. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.57 Let C be a circle with centre 'O' and HK is the chord of contact of pair of the tangents from point A. OA intersects the circle C at P and Q and B is the midpoint of HK, then Statement- : AB is the harmonic mean of AP and AQ. Statement- : AK is the Geometric mean of AB and AO and OA is the arithmetic mean of AP and AQ. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true.

7 Q.58 Statement-: Angle between the tangents drawn from the point P(3, 6) to the circle S : x + y 6x + 8y 75 = 0 is 90. Statement-: Point P lies on the director circle of S. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.59 Let C denotes a family of circles with centre on x-axis and touching the y-axis at the origin. and C denotes a family of circles with centre on y-axis and touching the x-axis at the origin. Statement-: Every member of C intersects any member of C at right angles at the point other than origin. Statement-: If two circles intersect at 90 at one point of their intersection, then they must intersect at 90 on the other point of intersection also. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.60 A circle is circumscribed about an equilateral triangle ABC and a point P on the minor arc joining A and B, is chosen. Let x = PA, y = PB and z = PC. (z is larger than both x and y.) Statement-: Each of the possibilities (x + y) greater than z, equal to z or less than z, is possible for some P. Statement-: In a triangle ABC, sum of the two sides of a triangle is greater than the third and the third side is greater than the difference of the two. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. Q.6 Statement-: Only one normal can be drawn through the point P(, 3) to the circle x + y 4x + 8y 6 = 0 Statement-: Passing through any point lying inside a given circle only one normal can be drawn. (A) Statement- is true, statement- is true and statement- is correct explanation for statement-. (B) Statement- is true, statement- is true and statement- is NOT the correct explanation for statement-. (C) Statement- is true, statement- is false. (D) Statement- is false, statement- is true. [COMPREHENSION TYPE] Paragraph for question nos. 6 to 64 Consider a circle x + y = 4 and a point P(4, ). 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.6 The value of lies in the interval (A) (0, 5 ) (B) (5, 30 ) (C) 30, 45 ) (D) (45, 60 ) Q.63 The intercept made by a tangent on the x-axis is (A) 9/4 (B) 0/4 (C) /4 (D) /4 Q.64 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 = 3 (D) x y = 4

8 Paragraph for question nos. 65 to 67 Let A, B, C be three sets of real numbers (x, y) defined as A : {(x, y): y } B : {(x, y): x + y 4x y 4 = 0} C : {(x, y): x + y = } Q.65 Number of elements in the A B C is (A) 0 (B) (C) (D) infinite Q.66 (x + ) + (y ) + (x 5) + (y ) has the value equal to (A) 6 (B) 5 (C) 36 (D) 49 Q.67 If the locus of the point of intersection of the pair of perpendicular tangents to the circle B is the curve S then the area enclosed between B and S is (A) 6 (B) 8 (C) 9 (D) 8 Paragraph for Question Nos. 68 to 70 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.68 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.69 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.70 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 (A) 0 (B) 8 (C) (D) None of these Paragraph for question nos. 7 to 73 Consider 3 circles S : x + y + x 3 = 0 S : x + y = 0 S 3 : x + y + y 3 = 0 Q.7 The radius of the circle which bisect the circumferences of the circles S = 0 ; S = 0 ; S 3 = 0 is (A) (B) (C) 3 (D) 0 Q.7 If the circle S = 0 is orthogonal to S = 0 ; S = 0 and S 3 = 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) 3 Q.73 The radius of the circle touching S = 0 and S = 0 at (, 0) and passing through (3, ) is (A) (B) (C) (D)

9 Paragraph for questions nos. 74 to 76 Consider the two quadratic polynomials x C a : y = ax a a and C : y = 4 4 Q.74 If the origin lies between the zeroes of the polynomial C a then the number of integral value(s) of 'a' is (A) (B) (C) 3 (D) more than 3 Q.75 If 'a' varies then the equation of the locus of the vertex of C a, is (A) x y 4 = 0 (B) x y 4 = 0 (C) x y + 4 = 0 (D) x + y 4 = 0 Q.76 For a = 3, if the lines y = m x + c and y = m x + c are common tangents to the graph of C a and C then the value of (m + m ) is equal to (A) 6 (B) 3 (C) / (D) none x Paragraph for Question Nos. 77 to 79 Consider a line pair ax + 3xy y 5x + 5y + c = 0 representing perpendicular lines intersecting each other at C and forming a triangle ABC with the x-axis. Q.77 If x and x are intercepts on the x-axis and y and y are the intercepts on the y-axis then the sum (x + x + y + y ) is equal to (A) 6 (B) 5 (C) 4 (D) 3 Q.78 Distance between the orthocentre and circumcentre of the triangle ABC is (A) (B) 3 (C) 7/4 (D) 9/4 Q.79 If the circle x + y 4y + k = 0 is orthogonal with the circumcircle of the triangle ABC then 'k' equals (A) / (B) (C) (D) 3/ [MULTIPLE OBJECTIVE TYPE] Q.80 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 + 3y = 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 Q.8 x x cos = y y sin = r, represents : (A) equation of a straight line, if is constant and r is variable (B) equation of a circle, if r is constant and is a variable (C) a straight line passing through a fixed point and having a known slope (D) a circle with a known centre and a given radius. Q.8 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) 4 Q.83 A family of linear functions is given by f (x) = + c(x + 3) where c R. If a member of this family meets a unit circle centred at origin in two coincident points then 'c' can be equal to (A) 3/4 (B) 0 (C) 3/4 (D) Q.84 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 : (A) (, 5) (B) (5, ) (C) ( 5, ) (D) (, 5)

11 ANSWER KEY CIRCLE Q. A Q. C Q.3 D Q.4 D Q.5 C Q.6 C Q.7 A Q.8 B Q.9 A Q.0 B Q. A Q. B Q.3 B Q.4 A Q.5 D Q.6 D Q.7 B Q.8 D Q.9 D Q.0 C Q. D Q. C Q.3 B Q.4 A Q.5 C Q.6 A Q.7 D Q.8 B Q.9 A Q.30 B Q.3 B Q.3 C Q.33 B Q.34 A Q.35 B Q.36 C Q.37 D Q.38 D Q.39 D Q.40 A Q.4 B Q.4 C Q.43 C Q.44 D Q.45 B Q.46 A Q.47 C Q.48 D Q.49 A Q.50 C Q.5 A Q.5 D Q.53 D Q.54 C Q.55 A Q.56 B Q.57 A Q.58 A Q.59 A Q.60 D Q.6 C Q.6 D Q.63 B Q.64 A Q.65 B Q.66 C Q.67 C Q.68 D Q.69 D Q.70 A Q.7 C Q.7 D Q.73 C Q.74 B Q.75 A Q.76 B Q.77 B Q.78 C Q.79 D Q.80 B,C Q.8 A,B,C,D Q.8 A,C,D Q.83 A,B Q.84 B,D Q.85 A,B,D Q.86 A,C Q.87 B,C,D Q.88 C,D Q.89 (A) P, Q, R; (B) Q, R; (C) Q, R, S; (D) P, S Q.90 (A) P, Q; (B) P, Q; (C) Q; (D) Q, S

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