Circle PYQs Set 3 (PDF)

Set 3 of 6 · Questions 61-90 of 164

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Circle · Set 3 (Q61-90)
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61. $Let C be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of \frac{2 π}{3} at its centre, is$ (1) $x^{2} + y^{2} = 1$ (2) $x^{2} + y^{2} = \frac{27}{4}$ (3) $x^{2} + y^{2} = \frac{9}{4}$ (4) $(x^{2} + y^{2} = \frac{3}{2})$ 62. $The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is :$ (1) $\frac{10}{3}$ (2) $\frac{3}{5}$ (3) $\frac{6}{5}$ (4) $\frac{5}{3}$ 63. $Let ABCD and AEFG be squares of side 4 and 2 units, respectively. The point E is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:$ (1) $r = 0$ (2) $2 r^{2} - 4 r + 1 = 0$ (3) $2 r^{2} - 8 r + 7 = 0$ (4) $r^{2} - 8 r + 8 = 0$ 64. $Let a line L pass through the point of intersection of the lines bx + 10y - 8 = 0 and 2x - 3y = 0, b \in R - {\frac{4}{3}} . If the line L also passes through the point (1, 1) and touches the circle 17 (x^{2} + y^{2}) =16, then the eccentricity of the ellipse \frac{x ^{2}}{5} + \frac{y ^{2}}{b ^{2}} = 1 is :$ (1) $\frac{2}{5}$ (2) $\frac{3}{5}$ (3) $\frac{1}{5}$ (4) $\frac{2}{5}$ 65. $A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle is:$ (1) $a hyperbola$ (2) $an ellipse$ (3) $a straight line$ (4) $a parabola$ 66. $Let A = {(x, y) \in R \times R ∣ 2 x^{2} + 2 y^{2} - 2x - 2y = 1} B = {(x, y) \in R \times R ∣ 4x+ + 4 y^{2} - 16y + 7 = 0} and C = {(x, y) \in R \times R ∣ x^{2} + y^{2} - 4x - 2y + 5 \leq r^{2}} Then the minimum value of \|r\| such that A \cup B \subseteq C is equal to$ (1) $\frac{3 + 10}{2}$ (2) $\frac{2 + 10}{2}$ (3) $\frac{3 + 2 5}{2}$ (4) $1 + 5$ 67. $If the length of the chord of the circle, x^{2} + y^{2} = r^{2} (r > 0) along the line, y - 2 x = 3 is r then r^{2} is equal to:$ (1) $\frac{9}{5}$ (2) $\frac{24}{5}$ (3) $\frac{12}{5}$ (4) $12$ 68. $The circle x^{2} + y^{2} = 4 x + 8 y + 5 intersects the line 3 x - 4 y = m at two distinct points if$ (1) $- 35 < m < 15$ (2) $15 < m < 65$ (3) $35 < m < 85$ (4) $- 85 < m < - 35$ 69. $The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4 x - 5 y = 20 to the circle x^{2} + y^{2} = 9 is$ (1) $20 (x^{2} + y^{2}) - 36 x + 45 y = 0$ (2) $20 (x^{2} + y^{2}) + 36 x - 45 y = 0$ (3) $36 (x^{2} + y^{2}) - 20 x + 45 y = 0$ (4) $36 (x^{2} + y^{2}) + 20 x - 45 y = 0$ 70. $The common tangent to the circles x^{2} + y^{2} = 4 and x^{2} + y^{2} + 6 x + 8 y - 24 = 0 also passes through the point:$ (1) $(4, -2)$ (2) $(- 6, 4)$ (3) $(6, - 2)$ (4) $(- 4, 6)$ 71. $The number of common tangents, to the circles x^{2} + y^{2} - 18x - I5y + 131 = 0 and x^{2} + y^{2} - 6x - 6y - 7 = 0, is$ (1) $3$ (2) $1$ (3) $4$ (4) $2$ 72. $If the four distinct points (4, 6), (- 1, 5), (0, 0) and (k, 3 k) lie on a circle of radius r, then 10 k + r^{2} is equal to$ (1) $34$ (2) $32$ (3) $35$ (4) $33$ 73. $If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is :$ (1) $(x^{2} + y^{2})^{2} = 4 R^{2} x^{2} y^{2}$ (2) $(x^{2} + y^{2})^{3} = 4 R^{2} x^{2} y^{2}$ (3) $(x^{2} + y^{2})^{2} = 4 R x^{2} y^{2}$ (4) $(x^{2} + y^{2}) (x + y) = R^{2} xy$ 74. $If a variable line, 3x + 4y - λ = 0 is such that the two circles x^{2} + y^{2} - 2x -2y + 1 = 0 and x^{2} + y^{2} -18x -2y +78 = 0 are on its opposite sides, then the set of all values of λ is the interval:$ (1) $(2, 17)$ (2) $[12, 21]$ (3) $[13, 23]$ (4) $(23,31)$ 75. $Let P be a point on the circle S : x^{2} + y^{2} = 4 with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N . Then, the mid-point of the line segment M N must lie on the curve$ (1) $(x + y)^{2} = 3xy$ (2) $x^{2/3} + y^{2/3} = 2^{4/3}$ (3) $x^{2} + y^{2} = 2xy$ (4) $x^{2} + y^{2} = x^{2} y^{2}$ 76. $Let the centre of a circle C be α, β and its radius r < 8. Let 3x + 4y = 24 and 3x-4y = 32 be two tangents and 4x + 3y = 1 be a normal to C. Then (α - β + r) is equal to$ (1) $7$ (2) $5$ (3) $6$ (4) $9$ 77. $If a circle C passing through at point (4, 0) touches the circle x^{2} + y^{2} + 4 x - 6 y = 12 externally at the point (1, - 1), then the radius of C is:$ (1) $25$ (2) $4$ (3) $5$ (4) $57$ 78. $Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point (3, 2) and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point (5, 5) is :$ (1) $22$ (2) $42$ (3) $4$ (4) $5$ 79. $Two circles with equal radii are intersecting at the points (0, 1) and (0, - 1) . The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is :$ (1) $1$ (2) $2$ (3) $22$ (4) $2$ 80. $A square is inscribed in the circle x^{2} + y^{2} - 10 x - 6 y + 30 = 0 . One side of this square is parallel to y = x + 3 . If (x_{i}, y_{i}) are the vertices of the square, then Σ (x_{i}^{2} + y_{i}^{2}) is equal to:$ (1) $148$ (2) $152$ (3) $160$ (4) $156$ 81. $The line x = y touches a circle at the point (1,1). If the circle also passes through the point (1, - 3), then its radius is:$ (1) $3$ (2) $22$ (3) $2$ (4) $32$ 82. $An angle of intersection of the curves, \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 and x^{2} + y^{2} = ab, a > b, is :$ (1) $tan^{- 1} (\frac{a + b}{ab})$ (2) $tan^{- 1} (\frac{a - b}{2 ab})$ (3) $tan^{- 1} (\frac{a - b}{ab})$ (4) $tan^{- 1} (2 ab)$ 83. $A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (α, β), then 3 β - 2 α is equal to:$ (1) $10$ (2) $15$ (3) $12$ (4) $14$ 84. $If a circle touches y -axis at (0, 4) and passes through (2, 0) then which of the following can-not be the tangent to the circle$ (1) $4 x - 3 y + 17 = 0$ (2) $3 x - 4 y - 24 = 0$ (3) $3 x + 4 y - 6 = 0$ (4) $4 x + 3 y - 8 = 0$ 85. $If the area of the triangle formed by the positive x - axis, the normal and the tangent to the circle (x - 2)^{2} + (y - 3)^{2} = 25 at the point (5, 7) is A, then 24 A is equal to$ 86. $The straight line 2x - 3y = 1 divides the circular region x2 + y2 \leq 6 into two parts. If S = {(2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})},$ 87. $For how many values of p, the circle x^{2} + y^{2} + 2x + 4y - p = 0 and the coordinate axes have exactly three common points?$ 88. $Let the abscissae of the two points P and Q be the roots of 2 x^{2} - rx + p = 0 and the ordinates of P and Q be the roots of x^{2} - sx - q = 0. If the equation of the circle described on PQ as diameter is 2(x^{2} + y^{2}) - 11x - 14y - 22 = 0, then 2r + s - 2q + p is equal to$ 89. $Let the centre of a circle, passing through the points (0, 0), (1, 0) and touching the circle x^{2} + y^{2} = 9, be (h, k) . Then for all possible values of the coordinates of the centre (h, k), 4 (h^{2} + k^{2}) is equal to_______$ 90. $The straight line 2 x - 3 y = 1 divides the circular region x^{2} + y^{2} \leq 6 into two parts. If S = ((2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, - \frac{1}{4}), (\frac{1}{8}, \frac{1}{4})), then the number of point(s) in S lying inside the smaller part is$