Hyperbola PYQs Set 4 (PDF)

Set 4 of 11 · Questions 91-120 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 4 (Q91-120)
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91. $If the circles x^{2} + y^{2} - 2 x - 2 y - 7 = 0 and x^{2} + y^{2} + 4 x + 2 y + k = 0 cut orthogonally, then the length of the common chord of the circles is$ (1) $\frac{12}{13}$ (2) $2$ (3) $5$ (4) $8$ 92. $Let P (x_{0}, y_{0}) be the point on the hyperbola 3 x^{2} - 4 y^{2} = 36 . which is nearest to the line 3 x + 2 y = 1 . Then 2 (y_{0} - x_{0}) is equal to$ (1) $- 3$ (2) $9$ (3) $- 9$ (4) $3$ 93. $The radius of the circle which cuts all the three circles x^{2} + y^{2} - 4 x - 4 y + 3 = 0, x^{2} + y^{2} + 4 x - 4 y + 3 = 0 and x^{2} + y^{2} + 4 x + 4 y + 3 = 0 orthogonally is$ (1) $1$ (2) $3$ (3) $5$ (4) $7$ 94. $If the vertices and foci of a hyperbola are respectively (\pm 3, 0) and (\pm 4, 0) then the parametric equations of that hyperbola are$ (1) $x = 3 sec θ, y = 7 tan θ$ (2) $x = 3 sec θ, y = 7 tan θ$ (3) $x = 3 sec θ, y = 7 tan θ$ (4) $x = 3 sec θ, y = 7 tan θ$ 95. $Let P be the point (1, 0) and Q be a point on the curve y^{2} = 8x. The locus of midpoint of PQ is$ (1) $x^{2} - 4y + 2 = 0$ (2) $x^{2} + 4y + 2 = 0$ (3) $y^{2} + 2x + 2 = 0$ (4) $y^{2} - 4x + 2 = 0$ 96. $If two tangents are drawn from the point P (\frac{π}{4}) on the circle x^{2} + y^{2} = 4 to the circle x^{2} + y^{2} = 1 then the slopes of the tangents are$ (1) $2 \pm 2$ (2) $1 \pm 2$ (3) $2 \pm 3$ (4) $1 \pm 3$ 97. $The locus of the point of intersection on the line 3 x - y - 43 k = 0 and 3 kx + ky - 43 = 0 for different real values of k is a hyperbola H . If e is the eccentricity of H then 4 e^{2} =$ (1) $48$ (2) $39$ (3) $13$ (4) $16$ 98. $If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:$ (1) $\frac{13}{12}$ (2) $2$ (3) $\frac{13}{6}$ (4) $\frac{13}{8}$ 99. $If x - 2 y + k = 0 is a tangent to the parabola y^{2} - 4 x - 4 y + 8 = 0, then the value of k is$ (1) $2$ (2) $\frac{2}{5}$ (3) $7$ (4) $-7$ 100. $Let e_{1} be the eccentricity of a hyperbola for which distance between its focii is 2 times the distance between its directrices and e_{2} be the eccentricity of another hyperbola for which the length of its transverse axis is twice the length of its the conjugate axis. Then e_{1} e_{2} =$ (1) $1$ (2) $\frac{10}{2}$ (3) $5$ (4) $\frac{5}{2}$ 101. $The image of the point (3, 4) with respect to the radical axis of the circles x^{2} + y^{2} + 8 x + 2 y + 10 = 0 and x^{2} + y^{2} + 7 x + 3 y + 10 = 0 is$ (1) $(3, 4)$ (2) $(- 4, - 3)$ (3) $(4, 3)$ (4) $(- 3, - 4)$ 102. $Find the equations of the tangents drawn to the circle x^{2} + y^{2} = 50 at the points where the line x + 7 = 0 meets it.$ (1) $7 x + y + 50 = 0 and 7 x - y + 50 = 0$ (2) $x + y = 0 and x - y = 0$ (3) $x + 7 y + 5 = 0 and y - 7 x + 5 = 0$ (4) $x + 7 y + 50 = 0 and x - 7 y + 50 = 0$ 103. $The centre of the circle passing through the point (1, 1) and orthogonal to the circles x^{2} + y^{2} + 3 x - 5 y + 7 = 0 and x^{2} + y^{2} - 6 x - 10 y + 9 = 0 is$ (1) $(- \frac{19}{52}, \frac{71}{52})$ (2) $(\frac{19}{52}, - \frac{55}{52})$ (3) $(- \frac{55}{52}, \frac{19}{52})$ (4) $(- \frac{19}{52}, \frac{55}{52})$ 104. $Let PQ be a chord of the hyperbola \frac{x ^{2}}{4} - \frac{y ^{2}}{b ^{2}} = 1, perpendicular to the x -axis such that OPQ is an equilateral triangle, O being the centre of the hyperbola. If the eccentricity of the hyperbola is 3, then the area of the triangle OPQ is$ (1) $23$ (2) $\frac{11}{5}$ (3) $\frac{9}{5}$ (4) $\frac{8 3}{5}$ 105. $If the common chord of the circles x^{2} + y^{2} + 4 y = 0 and x^{2} + y^{2} - 4 x - 5 = 0 is the diameter of the circle S = 0 then the abscissa of the centre of the circle S = 0 is$ (1) $\frac{- 13}{8}$ (2) $\frac{3}{8}$ (3) $\frac{3}{4}$ (4) $\frac{- 13}{4}$ 106. $The equation of a hyperbola whose asymptotes are 3 x \pm 5 y = 0 and vertices are (\pm 5, 0) is$ (1) $3 x^{2} - 5 y^{2} = 25$ (2) $5 x^{2} - 3 y^{2} = 225$ (3) $25 x^{2} - 9 y^{2} = 225$ (4) $9 x^{2} - 25 y^{2} = 225$ 107. $If the vertices of a hyperbola be at (- 2, 0) and (2, 0) and one of its foci be at (- 3, 0), then which one of the following points does not lie on this hyperbola?$ (1) $(- 6, 210)$ (2) $(26, 5)$ (3) $(4, 15)$ (4) $(6, 52)$ 108. $Let e be the eccentricity of the ellipse \frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1 . If \frac{1}{e} is the eccentricity of a hyperbola, then the eccentricity of its conjugate hyperbola is$ (1) $\frac{4}{3}$ (2) $\frac{3}{5}$ (3) $\frac{4}{5}$ (4) $\frac{3}{2}$ 109. $If L_{1} represents the radical axis of circles x^{2} + y^{2} - 4 x - 6 y + 5 = 0 and x^{2} + y^{2} - 2 x - 4 y - 1 = 0 and L_{2} represents the radical axis of x^{2} + y^{2} + 2 x + 2 y - 7 = 0 and x^{2} + y^{2} + x + y + 9 = 0, then$ (1) $L_{1} is parallel to L_{2} .$ (2) $L_{1} is perpendicular to L_{2} .$ (3) $L_{1} and L_{2} intersect at an angle 30^{\circ} .$ (4) $L_{1} and L_{2} intersect at (1, 7) .$ 110. $Let P (10, 215) be a point on the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, whose foci are S and S^{’} . If the length of its latus rectum is 8, then the square of the area of Δ PSS^{’} is equal to:$ (1) $900$ (2) $4200$ (3) $1462$ (4) $2700$ 111. $Equation of a common tangent to the parabola y^{2} = 4x and the hyperbola xy = 2 is$ (1) $x + 2y + 4 = 0$ (2) $x - 2y + 4 = 0$ (3) $4x + 2y + 1 = 0$ (4) $x + y + 1 = 0$ 112. $Two circles centered at (2, 3) and (5, 6) intersect each other. If the radii are equal, the equation of the common chord is$ (1) $x - y + 1 = 0$ (2) $x + y + 1 = 0$ (3) $x - y - 8 = 0$ (4) $x + y - 8 = 0$ 113. $The length of the latus rectum of 3 x^{2} - 4 y + 6 x - 3 = 0 is$ (1) $\frac{3}{4}$ (2) $\frac{4}{3}$ (3) $2$ (4) $3$ 114. $If x - y + 1 = 0 meets the circle x^{2} + y^{2} + y - 1 = 0 at A and B, then the equation of the circle with AB as diameter is$ (1) $2 (x^{2} + y^{2}) + 3 x - y + 1 = 0$ (2) $2 (x^{2} + y^{2}) + 3 x - y + 2 = 0$ (3) $2 (x^{2} + y^{2}) + 3 x - y + 3 = 0$ (4) $x^{2} + y^{2} + 3 x - y + 4 = 0$ 115. $If m_{1}, m_{2} are the slopes of the tangents drawn from a point (1, - 3) to the circle x^{2} + y^{2} - 6 x + 4 y + 12 = 0 then 9 (m_{1}^{2} + m_{2}^{2}) =$ (1) $16$ (2) $25$ (3) $4$ (4) $1$ 116. $Let P (\frac{π}{4}), Q (\frac{5 π}{4}), R (\frac{3 π}{4}), T (\frac{7 π}{4}) be the points on the hyperbola x^{2} - 4 y^{2} - 4 = 0 in the parametric form. Then the area of the quadrilateral PQRT is (in square units)$ (1) $42$ (2) $162$ (3) $322$ (4) $82$ 117. $If the line y = mx + c is a common tangent to the hyperbola \frac{x ^{2}}{100} - \frac{y ^{2}}{64} = 1 and the circle x^{2} + y^{2} = 36 then which one of the following is true?$ (1) $5 m = 4$ (2) $8 m + 5 = 0$ (3) $c^{2} = 369$ (4) $4 c^{2} = 369$ 118. $If 2 x - ky + 3 = 0, 3 x - y + 1 = 0 are conjugate lines with respect to 5 x^{2} - 6 y^{2} = 15, then k =$ (1) $6$ (2) $4$ (3) $3$ (4) $2$ 119. $Find the equation of the circle which passes through origin and cuts off the intercepts -2 and 3 over the X and Y -axes respectively.$ (1) $x^{2} + y^{2} - 2 x + 8 y = 0$ (2) $2 x^{2} + y^{2} + 2 x - 3 y = 0$ (3) $x^{2} + y^{2} - 2 x - 8 y = 0$ (4) $x^{2} + y^{2} + 2 x - 3 y = 0$ 120. $If (h, k) is the centre of the circle which passes through the origin and cuts the circles x^{2} + y^{2} + 4 x + 6 y + 12 = 0 and x^{2} + y^{2} + 4 x - 6 y + 9 = 0 orthogonally, then k - 2 h =$ (1) $0$ (2) $1$ (3) $-1$ (4) $5$