Circle PYQs Set 4 (PDF)

Set 4 of 6 · Questions 91-120 of 164

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Circle · Set 4 (Q91-120)
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91. $The sides of a rectangle are given by x = \pm a and y = \pm b . The equation of the circle passing through the vertices of the rectangle is$ (1) $x^{2} + y^{2} = a^{2}$ (2) $x^{2} + y^{2} = a^{2} + b^{2}$ (3) $x^{2} + y^{2} = a^{2} - b^{2}$ (4) $(x - a)^{2} + (y - b)^{2} = a^{2} + b^{2}$ 92. $If an angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is :$ (1) $\frac{13}{5}$ (2) $\frac{120}{13}$ (3) $\frac{60}{13}$ (4) $\frac{13}{2}$ 93. $Let C_{1} and C_{2} be the centres of the circles x^{2} + y^{2} - 2 x - 2 y - 2 = 0 and x^{2} + y^{2} - 6 x - 6 y + 14 = 0 respectively. If P and Q are the points of intersection of these circles then, the area (in sq. units) of the quadrilateral P C_{1} Q C_{2} is :$ (1) $8$ (2) $6$ (3) $9$ (4) $4$ 94. $The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point$ (1) $(-5, 2)$ (2) $(2, - 5)$ (3) $(5, - 2)$ (4) $(-2, 5)$ 95. $Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the arc AC, divides the arc AC such that \frac{length of arc AB}{length of arc BC} = \frac{1}{5}, and OC = α OA + β OB, then α + 2 (3 - 1) β is equal to$ (1) $23$ (2) $2 - 3$ (3) $53$ (4) $2 + 3$ 96. $Two tangents to the circle x^{2} + y^{2} = 4 at the points A and B meet at the point P (- 4, 0) . Then the area of the quadrilateral PAOB, O being the origin, is$ (1) $23 sq. units$ (2) $83 sq. units$ (3) $43 sq. units$ (4) $63 sq. units$ 97. $Let A (1, 4) and B (1, - 5) be two points. Let P be a point on the circle (x - 1)^{2} + (y - 1)^{2} = 1 such that (PA)^{2} + (PB)^{2} have maximum value, then the points, P, A and B lie on :$ (1) $a parabola$ (2) $a straight line$ (3) $a hyperbola$ (4) $an ellipse$ 98. $Let C be a circle with radius 10 units and centre at the origin. Let the line x + y = 2 intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope -1. Then, a distance (in units) between the chord PQ and the chord MN is$ (1) $3 - 2$ (2) $2 + 1$ (3) $2 - 1$ (4) $2 - 3$ 99. $A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are m and n, respectively, then m + n^{2} is equal to$ (1) $408$ (2) $414$ (3) $396$ (4) $312$ 100. $The set of all values of a^{2} for which the line x + y = 0 bisects two distinct chords drawn from a point P (\frac{1 + a}{2}, \frac{1 - a}{2}) on the circle 2 x^{2} + 2 y^{2} - (1 + a) x - (1 - a) y = 0 is equal to$ (1) $(8, \infty)$ (2) $(4, \infty)$ (3) $(0, 4)$ (4) $(2, 12)$ 101. $Let the equation of the circle, which touches x -axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y -axis be x^{2} + y^{2} - αx + βy + γ = 0 . If the circle lies below x -axis, then the ordered pair (2 a, b^{2}) is equal to$ (1) $(γ, β^{2} - 4 α)$ (2) $(α, β^{2} + 4 γ)$ (3) $(γ, β^{2} + 4 α)$ (4) $(α, β^{2} - 4 γ)$ 102. $If one of the diameters of the circle x^{2} + y^{2} - 10 x + 4 y + 13 = 0 is a chord of another circle C, whose center is the point of intersection of the lines 2 x + 3 y = 12 and 3 x - 2 y = 5, then the radius of the circle C is$ (1) $20$ (2) $4$ (3) $6$ (4) $32$ 103. $If the area of an equilateral triangle inscribed in the circle, x^{2} + y^{2} + 10 x + 12 y + c = 0 is 273 sq. units then c is equal is :$ (1) $13$ (2) $20$ (3) $-25$ (4) $25$ 104. $Let the circle S : 36 x^{2} + 36 y^{2} - 108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x - 2y = 4 and 2x - y = 5 lies inside the circle S, then :$ (1) $\frac{25}{9} < C < \frac{13}{3}$ (2) $100 < C < 165$ (3) $81 < C < 156$ (4) $100 < C < 156$ 105. $Let the locus of the mid points of the chords of circle x^{2} + (y - 1)^{2} = 1 drawn from the origin intersect the line x + y = 1 at P and Q . Then, the length of PQ is :$ (1) $\frac{1}{2}$ (2) $2$ (3) $\frac{1}{2}$ (4) $1$ 106. $Let the line x + y = 1 meet the circle x^{2} + y^{2} = 4 at the points A and B . If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to :$ (1) $14$ (2) $37$ (3) $214$ (4) $57$ 107. $The minimum distance between any two points P_{1} and P_{2} while considering point P_{1} on one circle and point P_{2} on the other circle for the given circles' equations x^{2} + y^{2} - 10 x - 10 y + 41 = 0 x^{2} + y^{2} - 24 x - 10 y + 160 = 0 is$ 108. $If the variable line 3x + 4y = α lies between the two circles (x - 1)^{2} + (y - 1)^{2} = 1 and (x - 9)^{2} + (y - 1)^{2} = 4 without intercepting a chord on either circle, then the sum of all the integral values of α is .$ 109. $Consider a circle (x - α)^{2} + (y - β)^{2} = 50, where α, β > 0 . If the circle touches the line y + x = 0 at the point P, whose distance from the origin is 42, then (α + β)^{2} is equal to$ 110. $If P is a point on the circle x^{2} + y^{2} = 4, Q is a point on the straight line 5 x + y + 2 = 0 and x - y + 1 = 0 is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such points P is ___.$ 111. $The diameter of the circle, whose centre lies on the line x + y = 2 in the first quadrant and which touches both the lines x = 3 and y = 2, is$ 112. $The number of integral values of k for which the line, 3 x + 4 y = k intersects the circle, x^{2} + y^{2} - 2 x - 4 y + 4 = 0 at two distinct points is$ 113. $Two circles each of radius 5 units touch each other at the point (1,2). If the equation of their common tangent is 4x+3y=10, and C_{1} (α, β) and C_{2} (γ, δ), C_{1} \neq = C_{2} are their centers, then ∣ (α + β) (γ + δ) ∣ is equal to$ 114. $A circle passing through the point (α, β) in the first quadrant touches the two coordinate axes at the points A and B. The point P is above the line AB. The point Q on the line segment AB is the foot of perpendicular from P on AB. If PQ is equal to 11 units, then the value of α β is________$ 115. $Two tangents PQ and PR drawn to the circle x^{2} + y^{2} - 2 x - 4 y - 20 = 0 from point P (16, 7) . If the center of the circle is C, then the area of quadrilateral PQCR will be$ 116. $Points P (- 3, 2), Q (9, 10) and R (α, 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2 x - ky = 1, then k is equal to _______$ 117. $The centres of two circles C_{1} and C_{2} each of unit radius are at a distance of 6 units from each other. Let P be the mid point of the line segment joining the centres of C_{1} and C_{2} and C be a circle touching circles C_{1} and C_{2} externally. If a common tangent to C_{1} and C passing through P is also a common tangent to C_{2} and C, then the radius of the circle C is$ 118. $A variable circle passes through the fixed point A (p, q) and touches x - axis. The locus of the other end of the diameter through A is (x - p)^{2} = Kqy Find K .$ 119. $Two circles in the first quadrant of radii r_{1} and r_{2} touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line x + y = 2. Then r_{1}^{2} + r_{2}^{2} - r_{1} r_{2} is equal to____.$ 120. $Equation of two diameters of a circle are 2 x - 3 y = 5 and 3 x - 4 y = 7 . The line joining the points (- \frac{22}{7}, - 4) and (- \frac{1}{7}, 3) intersects the circle at only one point P (α, β) . Then 17 β - α is equal to$