Hyperbola PYQs Set 5 (PDF)

Set 5 of 11 · Questions 121-150 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 5 (Q121-150)
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121. $The length of the chord joining the points (4 cos θ, 4 sin θ) and (4 cos (θ + 60^{\circ}), 4 sin (θ +) (60^{\circ})) of the circle x^{2} + y^{2} = 16 is$ (1) $16$ (2) $2$ (3) $4$ (4) $8$ 122. $If e_{1} is the eccentricity of the hyperbola x = sec θ, y = 2 tan θ and e_{2} is the eccentricity of the hyperbola x = 2 sec θ, y = tan θ, then \frac{e _{2}^{2}}{ϵ _{1}^{2}} =$ (1) $1$ (2) $2$ (3) $\frac{1}{2}$ (4) $\frac{1}{4}$ 123. $The equation of the circle which touches the circle x^{2} + y^{2} - 6 x + 6 y + 17 = 0 externally and having the lines x^{2} - 3 xy - 3 x + 9 y = 0 as two normals, is$ (1) $x^{2} + y^{2} - 2 x + 5 y - 1 = 0$ (2) $x^{2} + y^{2} + 2 x + 3 y + 1 = 0$ (3) $x^{2} + y^{2} - 6 x - 2 y + 1 = 0$ (4) $x^{2} + y^{2} + 4 x - 3 y + 3 = 0$ 124. $If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the tangent to the hyperbola at (4, 6) is:$ (1) $x - 2 y + 8 = 0$ (2) $2 x - 3 y + 10 = 0$ (3) $2 x - y - 2 = 0$ (4) $3 x - 2 y = 0$ 125. $Suppose the new axes X, Y are generated by rotating the coordinate axes x, y about the origin through an angle of 30^{\circ} in the anti-clockwise direction. Then, the transformed equation of x^{2} + 23 xy - y^{2} = 2 a^{2} with respect to new axes X, Y is$ (1) $X^{2} - Y^{2} = a^{2}$ (2) $X^{2} + Y^{2} = 2 a^{2}$ (3) $X^{2} + 23 XY - Y^{2} = 2 a^{2}$ (4) $X^{2} - Y^{2} = 2 a^{2}$ 126. $The length of the common chord of the circles x^{2} + y^{2} - 6 x - 4 y + 9 = 0 and x^{2} + y^{2} - 8 x - 6 y + 23 = 0 is$ (1) $2$ (2) $2$ (3) $22$ (4) $4$ 127. $Let the domain of the function f (x) = log_{3} log_{5} log_{7} (9 x - x^{2} - 13) be the interval (m, n) . Let the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 have eccentricity \frac{n}{3} and the length of the latus rectum \frac{8 m}{3} . Then b^{2} - a^{2} is equal to:$ (1) $7$ (2) $11$ (3) $5$ (4) $9$ 128. $If the line αx + 2 y = 1, where α \in R, does not meet the hyperbola x^{2} - 9 y^{2} = 9, then a possible value of α is:$ (1) $0.5$ (2) $0.6$ (3) $0.7$ (4) $0.8$ 129. $If the distance between the foci and the distance between the directrices of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 are in the ratio 3 : 2, then a : b is$ (1) $2 : 1$ (2) $3 : 2$ (3) $1 : 2$ (4) $2 : 1$ 130. $A circle S cuts three circles x^{2} + y^{2} - 4 x - 2 y + 4 = 0, x^{2} + y^{2} - 2 x - 4 y + 1 = 0 and x^{2} + y^{2} + 4 x + 2 y + 1 = 0 orthogonally. Then, the radius of S is$ (1) $\frac{29}{8}$ (2) $\frac{28}{11}$ (3) $\frac{29}{7}$ (4) $\frac{29}{5}$ 131. $The equation of the tangent to the parabola y^{2} = 4 x inclined at an angle of \frac{π}{4} to the positive direction of x -axis is$ (1) $x + y - 4 = 0$ (2) $x - y + 4 = 0$ (3) $x - y - 1 = 0$ (4) $x - y + 1 = 0$ 132. $The abscissae of two points P, Q are the roots of the equation 2 x^{2} + 4 x - 7 = 0 and their ordinates are the roots of the equation 3 x^{2} - 12 x - 1 = 0 . Then the centre of the circle with PQ as a diameter is$ (1) $(- 1, 2)$ (2) $(- 2, 6)$ (3) $(1, - 2)$ (4) $(2, - 6)$ 133. $If θ denotes the acute angle between the curves, y = 10 - x^{2} and y = 2 + x^{2} at a point of their intersection, then \| tan θ \| is equal to:$ (1) $\frac{4}{9}$ (2) $\frac{8}{15}$ (3) $\frac{7}{17}$ (4) $\frac{8}{17}$ 134. $A line parallel to the straight line 2 x - y = 0 is tangent to the hyperbola \frac{x ^{2}}{4} - \frac{y ^{2}}{2} = 1 at the point (x_{1}, y_{1}) . Then x_{1}^{2} + 5 y_{1}^{2} is equal to$ (1) $8$ (2) $6$ (3) $10$ (4) $5$ 135. $If A (1, 1), B (- 1, 1) and C (- 1, - 1) are three points and a point P moves such that PA^{2} = PB^{2} + PC^{2}, then the equation of the locus of P is$ (1) $x^{2} + y^{2} - 6 x - 2 y + 2 = 0$ (2) $x^{2} + y^{2} + 6 x + 2 y + 2 = 0$ (3) $x^{2} + y^{2} + 6 x - 2 y + 2 = 0$ (4) $x^{2} + y^{2} + 6 x + 2 y - 2 = 0$ 136. $If the angle between the circles x^{2} + y^{2} - 4 x - 6 y + k = 0 and x^{2} + y^{2} + 8 x - 4 y + 11 = 0 is \frac{π}{3}, then the value of k is$ (1) $-3$ (2) $36$ (3) $3$ (4) $2$ 137. $The radical centre of the circles x^{2} + y^{2} = 1, x^{2} + y^{2} - 2 x - 3 = 0 and x^{2} + y^{2} - 2 y - 3 = 0 is$ (1) $(1, 1)$ (2) $(1, - 1)$ (3) $(- 1, 1)$ (4) $(- 1, - 1)$ 138. $Consider the circles S_{1} : x^{2} + y^{2} + 2 x + 8 y - 23 = 0 and S_{2} : x^{2} + y^{2} - 4 x + 10 y + 19 = 0 If the polars of the centre of a circle with respect to the another circle are L_{1} and L_{2}, then L_{1}, L_{2} are$ (1) $Parallel and separated by a distance of 410 units$ (2) $Perpendicular and intersect at (1, 3)$ (3) $Perpendicular and intersect at (1, - 5)$ (4) $Parallel and separated by a distance of 210 units.$ 139. $Let one focus of the hyperbola H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 be at (10, 0) and the corresponding directrix be x = \frac{9}{10} . If e and l respectively are the eccentricity and the length of the latus rectum of H, then 9 (e^{2} + l) is equal to:$ (1) $12$ (2) $14$ (3) $15$ (4) $16$ 140. $The locus of the mid-points of the chords of the circle x^{2} - 2 x + y^{2} = 0 drawn from a point (0, 0) on it is$ (1) $x^{2} + y^{2} - x = 0$ (2) $2 x^{2} + y - 2 = 0$ (3) $y^{2} + x - 1 = 0$ (4) $x^{2} + 2 x + y - 3 = 0$ 141. $The transformed equation 3 x^{2} + 3 y^{2} + 2 xy = 2, when the coordinate axes are rotated through an angle 45^{\circ} is$ (1) $x^{2} + 2 y^{2} = 1$ (2) $2 x^{2} + y^{2} = 1$ (3) $x^{2} + y^{2} = 1$ (4) $x^{2} + 3 y^{2} = 1$ 142. $If the line x + y + k = 0 is a normal to the hyperbola \frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1 then k =$ (1) $\pm \frac{5}{13}$ (2) $\pm \frac{13}{5}$ (3) $\pm \frac{13}{5}$ (4) $\pm \frac{5}{13}$ 143. $The tangent to the parabola y^{2} = 4 x at the point where it intersects the circle x^{2} + y^{2} = 5 in the first quadrant, passes through the point:$ (1) $(- \frac{1}{3}, \frac{4}{3})$ (2) $(\frac{1}{4}, \frac{3}{4})$ (3) $(\frac{3}{4}, \frac{7}{4})$ (4) $(- \frac{1}{4}, \frac{1}{2})$ 144. $Let e_{1} and e_{2} be the eccentricities of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{b ^{2}} = 1 (b < 5) and the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1 respectively satisfying e_{1} e_{2} = 1 . If α and β are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair (α, β) is equal to :$ (1) $(8, 10)$ (2) $(\frac{24}{5}, 10)$ (3) $(\frac{20}{3}, 12)$ (4) $(8, 12)$ 145. $The equation of a circle with centre (5, 4) and touch the Y -axis is$ (1) $x^{2} + y^{2} - 10 x - 8 y - 16 = 0$ (2) $x^{2} + y^{2} - 10 x - 8 y + 9 = 0$ (3) $x^{2} + y^{2} + 10 x + 8 y + 16 = 0$ (4) $x^{2} + y^{2} - 10 x - 8 y + 16 = 0$ 146. $The radical centre of the circles x^{2} + y^{2} - 4 x - 6 y + 5 = 0, x^{2} + y^{2} - 2 x - 4 y - 1 = 0 and x^{2} + y^{2} - 6 x - 2 y = 0 = 0 lies on the line$ (1) $x + y - 5 = 0$ (2) $2 x - 4 y + 7 = 0$ (3) $4 x - 6 y + 5 = 0$ (4) $18 x - 12 y + 1 = 0$ 147. $If the focii of \frac{x ^{2}}{16} + \frac{y ^{2}}{4} = 1 and \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{3} = 1 coincide, then value of a is$ (1) $3$ (2) $\frac{1}{3}$ (3) $2$ (4) $3$ 148. $Locus of a point which moves such that its distance from the X -axis is twice its distance from the line x - y = 0 is$ (1) $x^{2} + 4 xy - y^{2} = 0$ (2) $2 x^{2} - 4 xy + y^{2} = 0$ (3) $x^{2} - 4 xy + y^{2} = 0$ (4) $x^{2} - 4 xy - y^{2} = 0$ 149. $The equation of the smallest circle passing through the intersection of the line x + y = 1 and the circle x^{2} + y^{2} = 9 is$ (1) $x^{2} + y^{2} - 9 - (x + y + 1) = 0$ (2) $x^{2} + y^{2} - 9 - (x + y - 1) = 0$ (3) $x^{2} + y^{2} - 9 - x + y - 1 = 0$ (4) $x^{2} + y^{2} - 9 + x + y - 1 = 0$ 150. $For all real values of k, the polar of the point (2 k, k - 4) with respect to x^{2} + y^{2} - 4 x - 6 y + 1 = 0 passes through the point$ (1) $(1, 1)$ (2) $(1, - 1)$ (3) $(- 3, 1)$ (4) $(3, 1)$