## Integer Type/Fill in The Blanks

Q.1₂₀₂₀ Let O be the centre of the circle x² + y² = r² , where r > √5/2. Suppose PQ is a chord of this circle and the equation of the line passing through P and Q is 2x + 4y = 5. If the centre of the circumcircle of the triangle OPQ lies on the line x + 2y = 4, then the value of r is…

Solution:
Let C be the centre of the circumcircle of the triangle OPQ ⇒ C lies on the line x + 2y = 4
Let R(h, k) be any point on the circumcircle of the triangle OPQ, such that O, C, and R are collinear
⇒ Equation of the line PQ: hx + ky = r², But equation of the line PQ is given 2x + 4y = 5
⇒ h/2 = k/4 = r²/5 ⇒ h = 2r²/5, k = 4r²/5
⇒ C = (r²/5, 2r²/5) and C lies on the line x + 2y = 4
⇒ (r²/5) + 2(2r²/5) = 4 ⇒ r² = 4 ⇒ r = 2

Q.2₂₀₁₇ For how many values of P the circle x² + y² + 2x + 4y – P = 0 and the co-ordinate axes have exactly three common points…

Solution:
Let the circle x² + y² + 2x + 4y – P = 0 ⇒ (x + 1)² + (y + 2)² = P + 5
When circle touches x-axis P + 5 = 4 ⇒ P = -1
When circle passes through the origin P + 5 = 5 ⇒ P = 0
Hence two values -1 and 0 of P are possible

Q.3₂₀₁₀ Two parallel chords of a circle of radius 2 are at a distance of (1 + √3) apart. If the chords subtend at the centre angles of π/k and 2π/k, where k > 0, then the value of [k] (where [k] is largest integer less than or equal to k ) is…

Solution:
2Cos(π/2k) + 2Cos(π/k) = 1 + √3 ⇒ Cos(π/2k) + Cos(π/k) = (1 + √3)/2
⇒ Cos(π/2k) + 2Cos²(π/2k) -1 = (1 + √3)/2
⇒ 4Cos(π/2k) = -1 ± (1 + 2√3) ⇒ Cos(π/2k) = (-2 – 2√3)/4, √3/2
Since, Cos(π/2k) ∈ [-1, 1] ⇒ Cos(π/2k) = √3/2
⇒ π/2k = π/6 ⇒ k = 3

Q.4₂₀₀₉ The centers of two circles C1 and C2 each of unit radius are at a distance of 6 units from each other. Let P be the midpoint of the line segment joining the centers of C1 and C2 and C be circle touching circles C1 and C2 externally. If a common tangent to C1 and C passing through P is also a common tangent to C2 and C, then the radius of the circle C is…

Solution:
Let r be the radius of the circle C
⇒ (r + 1)² – (r – 1)² = (4√2)² ⇒ r = 8

Q.5₁₉₉₇ For each natural number k let Cₖ denotes the circle with radius k centimetres and centre at the origin. On the circle Cₖ a particle moves k centimetres in the counter clockwise direction. After completing its motion on Cₖ the particle moves to Cₖ₊₁ in the radial direction. The motion of the particle continues in this manner. The particle starts at (1, 0). If the particle crosses the positive direction of the x-axis for the first time on the circle Cₙ, then n is equal to…

Solution:
The particle crosses the positive direction of the x-axis for the first time on the circle C₇
⇒ The value of n is 7

Q.6₁₉₉₆ The intercept on the line y = x by the circle x² + y² – 2x = 0 is AB. Equation of the circle with AB as a diameter is…

Solution:
The points of intersection of the line y = x and the circle x² + y² – 2x = 0 are A(0, 0) and B(1, 1)
⇒ The equation of the circle with A, B as the end points of diameter will be
x(x – 1) + y(y – 1) = 0 ⇒ x² + y² – x -y = 0

Q.7₁₉₉₄ A circle is inscribed in an equilateral triangle of side ‘a’. The area of any square inscribed in this circle is…

Solution:
Let the midpoint of the base of this equilateral triangle be origin and the sides of the equilateral triangle are given ‘a’
⇒ the radius of the circle is √3a/6
⇒ the area of the square is (diagonal)²/2 = (√3a/3)²/2 = 3a²/18 = a²/6

Q.8₁₉₉₃ The equation of the locus of the mid points of the chords of the circle 4x² + 4y² – 12x + 4y + 1 = 0 that subtend an angle of 2π/3 at its Centre is…

Solution:
Let AB be the chords of the given circle and C be the centre of the circle, such that C = (3/2, -(1/2))
Let P(h, k) be the mid points of the chords of the circle
⇒ AC = BC = radius = √(9/4) = 3/2
⇒ (√(h – 3/2)² + (k + 1/2)²)/(3/2) = 1/2
⇒ (x – 3/2)² + (y + 1/2)² = (9/16)

Q.9₁₉₉₃ If a circle passes through the points of intersection of the co-ordinate axes with the lines ax – y + 1 = 0 and x – 2y + 3 = 0, then these lines pass through a fixed point whose coordinates are…

Solution:
The equation of the family of the circle can be given by (ax – y + 1)(x – 2y + 3) + kxy = 0
The above equation will represent a circle if coefficient x² = coefficient y² and coefficient xy = 0
⇒ a = 2 and k = 5
⇒ these lines 2x – y + 1 = 0 and x – 2y + 3 = 0 pass through a fixed point (1/3, 5/3)

Q.10₁₉₈₈ If the circle C1: x² + y² = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has slope equal to 3/4, then coordinates of the centre of C2 are…

Solution:
Let the length of the common chord is the diameter of the circle C1: x² + y² = 16
⇒ the length of the common chord is 4
Let O and A are the centres of the circles C1 and C2 respectively
⇒ slope of AB = 3/4 ⇒ slope of OA = -4/3 {where B is the point of intesection of two circles}
⇒ equation of OA is x/(-3/5) = y/(4/5) = ± 3
⇒ centres are (-9/5, 12/5) or (9/5, -12/5)

## True / False Type

Q. 1 The straight line x + 3y = 0 is a diameter of the circle x² + y² – 6x + 2y = 0. {IIT JEE 1989, True/False type}.

Q.2 No tangent can be drawn from the point (5/2, 1) to the circumcircle of the triangle with vertices (1, √3), (1, -√3), and (3, -√3).{IIT JEE 1985, True/False type}.

## Multiple Choice Single Correct Type

Q.1 The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x² + y² = 9 is {IIT JEE 2012, Multiple Choice Single Correct Type}.

(A) 20(x² + y²) – 36x + 45y = 0
(B) 20(x² + y²) + 36x – 45y = 0
(C) 36(x² + y²) – 20x + 45y = 0
(D) 36(x² + y²) – 20x – 45y = 0

Q.2 The circle passing through the point (-1, 0) and touching the y-axis at (0, 2) also passes through the point. {IIT JEE 2011, Multiple Choice Single Correct Type}.

(A) (-3/2, 0)
(B) (-5/2, 2)
(C) (-3/2, 5/2)
(D) (-4, 0)

Q.3 Tangents drawn from the point P(1, 8) to the circle x² + y² – 6x – 4y – 11 = 0 touch the circle at the point A and B. The equation of the circumcircle of the triangle PAB is. {IIT JEE 2009, Multiple Choice Single Correct Type}.

(A) x² + y² + 4x – 6y + 19 = 0
(B) x² + y² – 4x – 10y + 19 = 0
(C) x² + y² – 2x + 6y – 29 = 0
(D) x² + y² – 6x – 4y + 19 = 0

Q.4 Let a and b be non-zero real numbers then the equation (ax² + by² + c) (x² + 6y² – 5xy) = 0 represents. {IIT JEE 2008, Multiple Choice Single Correct Type}.

(A) Four straight lines, when c = 0 and a, b are of the same sign.
(B) Two straight lines and a circle, when a = b and c is of sign opposite to that of a.
(C) Two straight lines and a hyperbola, when a and b are of the same sign and c is of the sign opposite to that of a.
(D) A circle and an ellipse, when a and b are of the same sign and c is of the sign opposite to that of a.

Q.5 Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is. {IIT JEE 2007, Multiple Choice Single Correct Type}.

(A) 3
(B) 2
(C) 3/2
(D) 1

Q.6 A circle is given by x² + (y – 1)² = 1, another circle C touches it externally and also the x-axis, then the locus of its center is. {IIT JEE 2005, Multiple Choice Single Correct Type}.

(A) {(x, y): x² = 4y} U {(x, y): y ≤ 0}
(B) {(x, y): x² + (y – 1)² = 4} U {(x, y): y ≤ 0}
(C) {(x, y): x² = y} U {(0, y): y ≤ 0}
(D) {(x, y): x² = 4y} U {(x, y): y ≤ 0}

Q.7 Tangent to the curve y = x² + 6 at a point P(1, 7) touches the circle x² + y² + 16x + 12y + c = 0 at a point Q, Then the co-ordinates of Q are {IIT JEE 2005, Multiple Choice Single Correct Type}.

(A) (-6, -11)
(B) (-9, -13)
(C) (-10, -15)
(D) (-6, -7)

Q.8 If one of the diameters of the circle x² + y² – 2x – 6y + 6 = 0 is a chord to the circle with center (2, 1), then the radius of the circle is {IIT JEE 2004, Multiple Choice Single Correct Type}.

(A) √3
(B) √2
(C) 3
(D) 2

Q.9 The centre of circle inscribed in square formed by the lines x² – 8x + 12 = 0 and y² – 14y + 45 = 0 is {IIT JEE 2003, Multiple Choice Single Correct Type}.

(A) (4, 7)
(B) (7, 4)
(C) (9, 4)
(D) (4, 9)

Q.10 If a > 2b > 0, then the positive value of ‘m’ for which y = mx – b√(1 +m²) is a common tangent to x² + y² = b² and (x – a)² + y² = b² is {IIT JEE 2002, Multiple Choice Single Correct Type}.

(A) 2b/√(a² – 4b²)
(B) √(a² – 4b²)/(2b)
(C) 2b/(a – 2b)
(D) b/(a – 2b)

Q.11 If the tangent at the point P on the circle x² + y² + 6x + 6y = 2 meets the straight line 5x – 2y + 6 = 0 at point Q on the y-axis, then the length of PQ is {IIT JEE 2002, Multiple Choice Single Correct Type}.

(A) 2√5
(B) 3√5
(C) 4
(D) 5

Q.12 Let AB be a chord of the circle x² + y² = r² subtending a right angle at the center, then the locus of the centroid of the triangle PAB as P moves on the circle is {IIT JEE 2001, Multiple Choice Single Correct Type}.

(A) A Parabola
(B) A Circle
(C) An Ellipse
(D) A Pair of Straight Lines

Q.13 Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point x on the circumference of the circle then 2r equals {IIT JEE 2001, Multiple Choice Single Correct Type}.

(A) √PQ.RS
(B) (PQ + RS)/2
(C) 2PQ.RS/(PQ + RS)
(D) √(PQ² + RS²)/2

Q.14 The triangle PQR is inscribed in the circle x² + y² = 25. If Q and R have coordinates (3, 4) and (-4, 3) respectively, then the angle of QPR is {IIT JEE 2000, Multiple Choice Single Correct Type}.

(A) pi/2
(B) pi/3
(C) pi/4
(D) pi/6

Q.15 If the circles x² + y² + 2x + 2ky + 6 = 0 and x² + y² + 2ky + k = 0 intersect orthogonally, the the value of ‘k’ is {IIT JEE 2000, Multiple Choice Single Correct Type}.

(A) 2 or -3/2
(B) -2 or -3/2
(C) 2 or 3/2
(D) -2 or 3/2

Q.16 If two distinct chords, drawn from the point (p, q) on the circle x² + y² – px – qy = 0 (where p, q ≠ 0) are bisected by the x-axis, then {IIT JEE 1999, Multiple Choice Single Correct Type}.

(A) p² = q²
(B) p² = 8q²
(C) p² < 8q²
(D) p² > 8q²

Q.17 The number of common tangents to the circles x² + y² = 4 and x² + y² – 6x – 8y = 24 is/are {IIT JEE 1998, Multiple Choice Single Correct Type}.

(A) 0
(B) 1
(C) 3
(D) 4

Q.18 The angle between a pair of tangents drawn from a point ‘P’ to the circle x² + y² + 4x – 6y + 9sin²a + 13cos²a = 0 is 2a. The equation of the locus of the point ‘P’ is {IIT JEE 1996, Multiple Choice Single Correct Type}.

(A) x² + y² + 4x – 6y + 4 = 0
(B) x² + y² + 4x – 6y – 9 = 0
(C) x² + y² + 4x – 6y – 4 = 0
(D) x² + y² + 4x – 6y + 9 = 0

Q.19 The locus of the center of a circle which touches externally the circle x² + y² – 6x – 6y + 14 = 0 and also touches the y-axis is given by the equation {IIT JEE 1993, Multiple Choice Single Correct Type}.

(A) x² – 6x – 10y + 14 = 0
(B) x² – 10x – 6y + 14 = 0
(C) y² – 6x – 10y + 14 = 0
(D) y² – 10x – 6y + 14 = 0

Q.20 The center of a circle passing through the points (0, 0), (1, 0) and touching the circle x² + y² = 9 is {IIT JEE 1992, Multiple Choice Single Correct Type}.

(A) (3/2, 1/2)
(B) (1/2, 3/2)
(C) (1/2, 1/2)
(D) (1/2, -√2)

Q.21 The lines 2x – 3y = 5 and 3x – 4y = 7 are diameters of a circle of area 154 square units. Then the equation of the circle is {IIT JEE 1989, Multiple Choice Single Correct Type}.

(A) x² + y² + 2x – 2y = 62
(B) x² + y² – 2x + 2y = 47
(C) x² + y² + 2x – 2y = 47
(D) x² + y² – 2x + 2y = 62