Circle PYQs Set 5 (PDF)

Set 5 of 6 · Questions 121-150 of 164

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Circle · Set 5 (Q121-150)
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121. $Let C_{1} be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C_{2} be the circle with centre (1, 3) that touches C_{1} externally at the point (α, β) . If (β - α)^{2} = \frac{m}{n}, gcd (m, n) = 1, then m + n is equal to$ (1) $22$ (2) $13$ (3) $9$ (4) $31$ 122. $Four distinct points (2 k, 3 k), (1, 0), (0, 1) and (0, 0) lie on a circle for k equal to :$ (1) $\frac{2}{13}$ (2) $\frac{3}{13}$ (3) $\frac{5}{13}$ (4) $\frac{1}{13}$ 123. $Choose the incorrect statement about the two circles whose equations are given below: x^{2} + y^{2} - 10 x - 10 y + 41 = 0 and x^{2} + y^{2} - 16 x - 10 y + 80 = 0$ (1) $Distance between two centres is the average of radii of both the circles.$ (2) $Both circles' centres lie inside region of one another.$ (3) $Both circles pass through the centre of each other.$ (4) $Circles have two intersection points.$ 124. $Let a circle C touch the lines L_{1} : 4x - 3y + K_{1} = 0 and L_{2} : 4x - 3y + K_{2} = 0, K_{1}, K_{2} \in R. If a line passing through the centre of the circle C intersects L_{1} at (-1, 2) and L_{2} at (3, - 6), then the equation of the circle C is$ (1) $(x - 1)^{2} + (y - 2)^{2} = 4$ (2) $(x + 1)^{2} + (y - 2)^{2} = 4$ (3) $(x - 1)^{2} + (y + 2)^{2} = 16$ (4) $(x - 1)+ + (y - 2)^{2} = 16$ 125. $Let r_{1} and r_{2} be the radii of the largest and smallest circles, respectively, which pass through the (-4,1) point and having their centres on the circumference of the circle x^{2} + y^{2} + 2x + 4y - 4 = 0 If \frac{r _{1}}{r _{2}} = a + b 2, then a+b is equal to :$ (1) $3$ (2) $11$ (3) $5$ (4) $7$ 126. $A circle touching the x-axis at (3, 0) and making an intercept of length 8 on the y-axis passes through the point:$ (1) $(3, 10)$ (2) $(3, 5)$ (3) $(2, 3)$ (4) $(1, 5)$ 127. $Let O be the origin and OP and OQ be the tangents to the circle x^{2} + y^{2} -6x + 4y + 8 = 0 at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point (α, \frac{1}{2}) then α value of a is$ (1) $\frac{3}{2}$ (2) $- \frac{1}{2}$ (3) $\frac{5}{2}$ (4) $1$ 128. $The sum of the squares of the lengths of the chords intercepted on the circle, x^{2} + y^{2} = 16, by the lines, x + y = n, n ϵ N, where N is the set of all natural numbers, is :$ (1) $320$ (2) $105$ (3) $160$ (4) $210$ 129. $The tangent and the normal lines at the point (3, 1) to the circle x^{2} + y^{2} = 4 and the x-axis form a triangle. The area of this triangle (in sq. units) is:$ (1) $\frac{4}{3}$ (2) $\frac{1}{3}$ (3) $\frac{2}{3}$ (4) $\frac{1}{3}$ 130. $Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3 x + 2 y + 2 = 0 . Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:$ (1) $3$ (2) $22$ (3) $23$ (4) $42$ 131. $Tangents drawn from the point P(1, 8) to the circle x2 + y2 - 6x - 4y - 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is$ (1) $x2 + y2 + 4x - 6y + 19 = 0$ (2) $x2 + y2 - 4x - 10y + 19 = 0$ (3) $x2 + y2 - 2x + 6y - 29 = 0$ (4) $x2 + y2 - 6x - 4y + 19 = 0$ 132. $A square is inscribed in the circle x^{2} + y^{2} - 6 x + 8 y - 103 = 0 with its sides parallel to the coordinate axes. Then the distance of the vertex of this square which is nearest to the origin is:$ (1) $6$ (2) $137$ (3) $41$ (4) $13$ 133. $Let C be the circle of minimum area touching the parabola y = 6 - x^{2} and the lines y = 3 \| x \| . Then, which one of the following points lies on the circle C ?$ (1) $(1, 2)$ (2) $(1, 1)$ (3) $(2, 2)$ (4) $(2, 4)$ 134. $Consider a family of circles which are passing through the point (-1, 1) and are tangent to x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval$ (1) $0 < k < 1 < 2$ (2) $k \geq ½$ (3) $- 1/2 \leq k \leq 1/2$ (4) $k \leq 1/2$ 135. $Three circles of radii a, b, c (a < b<c) touch each other externally. If they have x -axis as a common tangent, then:$ (1) $\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$ (2) $\frac{1}{b} = \frac{1}{a} + \frac{1}{c}$ (3) $a, b, c are in A.P.$ (4) $a, b, c are in A.P.$ 136. $The line 2 x - y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x - 2 y = 4 . Then, the radius of the circle is:$ (1) $35$ (2) $53$ (3) $54$ (4) $55$ 137. $In the circle given below, let OA = 1 unit, OB = 13 unit and PQ ⊥ OB . Then, the area of the triangle PQB (in square units) is:$ (1) $263$ (2) $242$ (3) $243$ (4) $262$ 138. $Let y = x + 2, 4 y = 3 x + 6 and 3 y = 4 x + 1 be three tangent lines to the circle (x - h)^{2} + (y - k)^{2} = r^{2} . Then h + k is equal to$ (1) $5$ (2) $5 (1 + 2)$ (3) $6$ (4) $52$ 139. $Let c and d be vectors such that c + d ∣ = 29 and c \times (2 i + 3 j + 4 k) = (2 i + 3 j + 4 k) \times d . If λ_{1}, λ_{2} (λ_{1} > λ_{2}) are the possible values of (c + d) \cdot (- 7 i + 2 j + 3 k), then the equation K^{2} x^{2} + (K^{2} - 5 K + λ_{1}) xy + (3 K + \frac{λ _{2}}{2}) y^{2} - 8 x + 12 y + λ_{2} = 0 represents a circle, for K equal to :$ (1) $1$ (2) $4$ (3) $-1$ (4) $2$ 140. $A circle C_{1} passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C_{1} Let C_{2} be the circle with OA as a diameter. If the tangent to C_{2} at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to :$ (1) $1 : 4$ (2) $1 : 5$ (3) $2 : 5$ 141. $The locus of the centres of the circles, which touch the circle, x^{2} + y^{2} = 1 externally, also touch the y-axis and lie in the first quadrant, is:$ (1) $x = 1 + 4 y, y \geq 0$ (2) $y = 1 + 2 x, x \geq 0$ (3) $y = 1 + 4 x, x \geq 0$ (4) $x = 1 + 2 y, y \geq 0$ 142. $If the radius of the largest circle with centre (2, 0) inscribed in the ellipse x^{2} + 4 y^{2} = 36 is r, then 12 r^{2} is equal to$ (1) $115$ (2) $92$ (3) $69$ (4) $72$ 143. $If a circle of constant radius 3 k passes through the origin and meets the axes at A and B, the locus of the centroid of the triangle OAB is the circle x^{2} + y^{2} = M k^{2} . Find M .$ 144. $Let the mirror image of a circle c_{1} : x^{2} + y^{2} - 2x -6y + α = 0 in line y = x + 1 be c_{2} : 5 x^{2} + 5 y^{2} + 10gx + 10fy + 38 = 0. If r is the radius of circle c_{2}, then α + 6 r^{2} is equal to________$ 145. $The absolute difference between the squares of the radii of the two circles passing through the point (- 9, 4) and touching the lines x + y = 3 and x - y = 3, is equal to _______.$ 146. $The locus of a point, which moves such that the sum of squares of its distances from the points (0,0), (1,0), (0,1), (1,1) is 18 units, is a circle of diameter d. Then d^{2} is equal to$ 147. $If one of the diameters o of the circle x^{2} + y^{2} - 2 x - 6 y + 6 = 0 is a chord of another circle^{’} C^{’} whose center is at (2, 1), then its radius is$ 148. $The equation of the circle whose center is (3,-1) and which cuts off a chord of length 6 on the line 2 x - 5 y + 18 = 0 is (x - 3)^{2} + (y + 1)^{2} = K . Find K .$ 149. $Let the line L : 2 x + y = α pass through the point of the intersection P (in the first quadrant) of the circle x^{2} + y^{2} = 3 and the parabola x^{2} = 2 y . Let the line L touch two circles C_{1} and C_{2} of equal radius 23 . If the centres Q_{1} and Q_{2} of the circles C_{1} and C_{2} lie on the y -axis, then the square of the area of the triangle P Q_{1} Q_{2} is equal to$ 150. $Let P (a_{1}, b_{1}) and Q (a_{2}, b_{2}) be two distinct points on a circle with centre C (2, 3) . Let O be the origin and O C be perpendicular to both C P and C Q . If the area of the triangle O C P is \frac{35}{2}, then a_{1}^{2} + b_{1}^{2} + a_{2}^{2} + b_{2}^{2} is equal to _____$