Hyperbola PYQs Set 6 (PDF)

Set 6 of 11 · Questions 151-180 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 6 (Q151-180)
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151. $If e_{1} is the eccentricity of the ellipse \frac{x ^{2}}{16} + \frac{y ^{2}}{25} = 1 and e_{2} is the eccentricity of a hyperbola passing through the foci of the given ellipse and e_{1} e_{2} = 1, then the equation of such a hyperbola among the following is$ (1) $\frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1$ (2) $\frac{y ^{2}}{9} - \frac{x ^{2}}{16} = 1$ (3) $\frac{x ^{2}}{9} - \frac{y ^{2}}{25} = 1$ (4) $\frac{x ^{2}}{25} - \frac{y ^{2}}{9} = 1$ 152. $If a circle of radius r touches the positive coordinate axes and also the circle x^{2} + y^{2} - 12 x - 10 y + 52 = 0 externally, then the distance between the centres of the two circles is$ (1) $7$ (2) $5$ (3) $6$ (4) $8$ 153. $The point on the circle x^{2} + y^{2} = 4 whose distance from the line 4 x + 3 y - 12 = 0 is 4/5 units is equal to$ (1) $(\frac{12}{25}, \frac{36}{25})$ (2) $(4, 0)$ (3) $(2, 0)$ (4) $(\frac{- 14}{25}, \frac{48}{25})$ 154. $If the line ax + y = c touches both the curves x^{2} + y^{2} = 1 and y^{2} = 4 2 x then \| c \| is equal to$ (1) $2$ (2) $\frac{1}{2}$ (3) $\frac{1}{2}$ (4) $2$ 155. $If the foci of a hyperbola are same as that of the ellipse \frac{x ^{2}}{9} + \frac{y ^{2}}{25} = 1 and the eccentricity of the hyperbola is \frac{15}{8} times the eccentricity of the ellipse, then the smaller focal distance of the point (2, \frac{14}{3} \frac{2}{5}) on the hyperbola, is equal to$ (1) $7 \frac{2}{5} - \frac{8}{3}$ (2) $14 \frac{2}{5} - \frac{4}{3}$ (3) $14 \frac{2}{5} - \frac{16}{3}$ (4) $7 \frac{2}{5} + \frac{8}{3}$ 156. $Let the centre of the circle S = 0 lie on the line x + y - 5 = 0 and also lie in the first quadrant. If this circle touches both the lines x - 2 = 0 and y - 5 = 0, then the area of the circle is$ (1) $π sq. units$ (2) $2 π sq. units$ (3) $4 π sq. units$ (4) $1 4 π sq. units$ 157. $The condition for the line y = mx + c to be a normal to the parabola y^{2} = 4 ax is$ (1) $c = - \frac{a}{m}$ (2) $c = - 2 am - a m^{3}$ (3) $c = 2 am + a m^{3}$ (4) $c = \frac{a}{m}$ 158. $If 2x - y + 1 = 0 is a tangent to the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{16} = 1, then which of the following CANNOT be sides of a right angled triangle ?$ (1) $a, 4, 1$ (2) $2a, 4, 1$ (3) $a, 4, 2$ (4) $2a, 8, 1$ 159. $Let x + y + 1 = 0 and x - y + 4 = 0 be the asymptotes of a hyperbola H . If (1, 1) is a point on H, then the length of the latus rectum of H is$ (1) $43$ (2) $3$ (3) $42$ (4) $5$ 160. $If P (\frac{π}{6}) is a point on the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, S, S^{'} are its foci and S P + S^{'} P = 2 (S P - S^{'} P), then e =$ (1) $2$ (2) $2$ (3) $3$ (4) $3$ 161. $The equation of the common tangent of the two touching circles, y^{2} + x^{2} - 6 x - 12 y + 37 = 0 and x^{2} + y^{2} - 6 y + 7 = 0 is$ (1) $x - y - 5 = 0$ (2) $x - y + 5 = 0$ (3) $x + y - 5 = 0$ (4) $x + y + 5 = 0 .$ 162. $If the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, (a > b) has eccentricity and the length of the latus rectum respectively equal to \frac{5}{4} and 9 , then ab is equal to$ (1) $122$ (2) $183$ (3) $48$ (4) $20$ 163. $Let \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b be an ellipse, whose eccentricity is \frac{1}{2} and the length of the latus rectum is 14 . Then the square of the eccentricity of \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 is :$ (1) $3$ (2) $7/2$ (3) $3/2$ (4) $5/2$ 164. $If the angle between the asymptotes of a hyperbola is 30^{\circ} then its eccentricity is$ (1) $5 - 2$ (2) $6 - 3$ (3) $5 - 3$ (4) $6 - 2$ 165. $Find the maximum distance of the point K (10, 7) from the circle x^{2} + y^{2} - 4 x - 2 y - 20 = 0$ (1) $25$ (2) $10$ (3) $15$ (4) $5$ 166. $A hyperbola having its centre at the origin is passing through the point (5, 2) and has transverse axis of length 8 along the X -axis. Then the eccentricity of its conjugate hyperbola is$ (1) $\frac{13}{3}$ (2) $\frac{13}{3}$ (3) $\frac{13}{2}$ (4) $\frac{13}{2}$ 167. $If e_{1} and e_{2} are eccentricities of the ellipse \frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1 and the hyperbola \frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1 respectively and if point (e_{1}, e_{2}) lies on ellipse 15 x^{2} + 3 y^{2} = k . Then find value of k .$ (1) $14$ (2) $15$ (3) $16$ (4) $17$ 168. $If the line 3 x - my + 5 = 0 is a tangent to the hyperbola 3 x^{2} - 4 y^{2} = 300 then the square of the Y -intercept made by this tangent line is$ (1) $\frac{25}{3}$ (2) $\frac{35}{3}$ (3) $\frac{45}{7}$ (4) $\frac{15}{7}$ 169. $If P is a point on the parabola y^{2} = 8 x and A is the point (1, 0), then the locus of the mid-point of the line segment AP is$ (1) $y^{2} = 4 (x - \frac{1}{2})$ (2) $y^{2} = 2 (2 x + 1)$ (3) $y^{2} = x - \frac{1}{2}$ (4) $y^{2} = 2 x + 1$ 170. $If x = t^{2} + 2 and y = 2 t represent the parametric equation of the parabola$ (1) $(x - 2)^{2} = 4 y$ (2) $x^{2} = 4 (y - 2)$ (3) $(y - 2)^{2} = 4 x$ (4) $y^{2} = 4 (x - 2)$ 171. $The transformed equation of 3 x^{2} - 4 xy = r^{2}, when the coordinate axes are rotated through an angle tan^{- 1} (2) is$ (1) $X^{2} - 4 Y^{2} = r^{2}$ (2) $2 XY + r^{2} = 0$ (3) $4 Y^{2} - X^{2} = r^{2}$ (4) $XY = r^{2}$ 172. $A circle passes through the centre of another circle x^{2} + y^{2} - 3 x - 4 y - 1 = 0 and whose centre is (5, 2) . Then the equation of this circle is ...........$ (1) $4 x^{2} + 4 y^{2} - 40 x - 16 y + 67 = 0$ (2) $3 x^{2} + 3 y^{2} - 40 x - 16 y + 67 = 0$ (3) $2 x^{2} + 2 y^{2} - 40 x - 16 y + 67 = 0$ (4) $x^{2} + y^{2} - 10 x - 4 y + 67 = 0$ 173. $Let L (x_{1}, 4) be the end of the Latus rectum of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 lying in the first quadrant and let S (8, y_{1}) be the focus of the given hyperbola. Then the length of its transverse axis is$ (1) $2 (17 - 1)$ (2) $4 (17 - 1)$ (3) $2 (17 + 1)$ (4) $4 (17 + 1)$ 174. $In order to eliminate the first degree terms from the equation 4 x^{2} + 8 xy + 10 y^{2} - 8 x - 44 y + 14 = 0 the point to which the origin has to be shifted is$ (1) $(- 2, 3)$ (2) $(2, - 3)$ (3) $(1, - 3)$ (4) $(- 1, 3)$ 175. $A circle is drawn touching the X -axis, with its centre at the point of reflection of (m, n) on the line y - x = 0 . Then the equation of the circle is$ (1) $x^{2} + y^{2} - 2 mx - 2 ny + m^{2} = 0$ (2) $x^{2} + y^{2} - 2 mx + 2 ny + m^{2} = 0$ (3) $x^{2} + y^{2} + 2 nx - 2 my - n^{2} = 0$ (4) $x^{2} + y^{2} - 2 nx - 2 my + n^{2} = 0$ 176. $Let H be the hyperbola, whose foci and (1 \pm 2, 0) and eccentricity is 2 . Then the length of its latus rectum is ________$ (1) $2$ (2) $3$ (3) $\frac{5}{2}$ (4) $\frac{3}{2}$ 177. $Consider a hyperbola H having centre at the origin and foci on the x -axis. Let C_{1} be the circle touching the hyperbola H and having the centre at the origin. Let C_{2} be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of C_{1} and C_{2} are 36 π and 4 π, respectively, then the length (in units) of latus rectum of H is$ (1) $\frac{14}{3}$ (2) $\frac{28}{3}$ (3) $\frac{11}{3}$ (4) $\frac{10}{3}$ 178. $The number of real circles cutting orthogonally the circle x^{2} + y^{2} + 2 x - 2 y + 7 = 0 is$ (1) $0$ (2) $1$ (3) $2$ (4) $infinitely many$ 179. $Let the transverse axis of a hyperbola H be parallel to the X - axis and x^{2} + y^{2} - 2 x - 4 y + 3 = 0 be the equation of the auxiliary circle of H . If the asymptotes of H are at right angles, then the equation of the hyperbola is$ (1) $3 x^{2} - 2 y^{2} - 6 x + 8 y - 11 = 0$ (2) $x^{2} - y^{2} + 2 x + 4 y - 5 = 0$ (3) $3 x^{2} - 2 y^{2} + 6 x + 8 y - 11 = 0$ (4) $x^{2} - y^{2} - 2 x + 4 y - 5 = 0$ 180. $If the lines x + y = a and x - y = b touch the curve y = x^{2} - 3 x + 2 at a points where the curve intersects the x -axis, then \frac{a}{b} is equal to:$