Hyperbola PYQs Set 7 (PDF)

Set 7 of 11 · Questions 181-210 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 7 (Q181-210)
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181. $Let H : \frac{- x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 be the hyperbola, whose eccentricity is 3 and the length of the latus rectum is 43 . Suppose the point (α, 6), α > 0 lies on H . If β is the product of the focal distances of the point (α, 6), then α^{2} + β is equal to$ (1) $172$ (2) $171$ (3) $169$ (4) $170$ 182. $The lines of the form x cos φ + y sin φ = P are chords of the hyperbola 4 x^{2} - y^{2} = 4 a^{2} which subtend a right angle at the centre of the hyperbola. If these chords touch a circle with centre at (0, 0), then the radius of that circle is$ (1) $\frac{2 a}{3}$ (2) $\frac{a}{3}$ (3) $2 a$ (4) $\frac{a}{2}$ 183. $A hyperbola having the transverse axis of length 2 has the same foci as that of the ellipse of 3 x^{2} + 4 y^{2} = 12 then this hyperbola does not pass through which of the following points?$ (1) $(- \frac{3}{2}, 1)$ (2) $(\frac{3}{2}, \frac{1}{2})$ (3) $(\frac{1}{2}, 0)$ (4) $(1, - \frac{1}{2})$ 184. $The locus of a variable point whose chord of contact w.r.t the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 subtends a right angle at the origin is$ (1) $\frac{x ^{2}}{4 a ^{2}} - \frac{y ^{2}}{4 b ^{2}} = 1$ (2) $(\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}}) = \frac{x ^{2}}{a ^{4}} + \frac{y ^{2}}{b ^{4}}$ (3) $\frac{x}{a} - \frac{y}{b} = \frac{1}{a ^{2}} + \frac{1}{b ^{2}}$ (4) $\frac{x ^{2}}{a ^{4}} + \frac{y ^{2}}{b ^{4}} = \frac{1}{a ^{2}} - \frac{1}{b ^{2}}$ 185. $The centre of a circle which cuts x^{2} + y^{2} + 6 x - 1 = 0, x^{2} + y^{2} - 3 y + 2 = 0 and x^{2} + y^{2} + x + y - 3 = 0 orthogonally is$ (1) $(\frac{1}{7}, \frac{9}{7})$ (2) $(\frac{- 1}{7}, \frac{- 9}{7})$ (3) $(\frac{1}{7}, \frac{- 9}{7})$ (4) $(\frac{- 1}{7}, \frac{9}{7})$ 186. $The locus of the point of intersection of the tangents at the endpoints of normal chords of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 is$ (1) $\frac{a ^{6}}{x ^{2}} + \frac{b ^{6}}{y ^{2}} = (a^{2} + b^{2})^{2}$ (2) $\frac{a ^{6}}{x ^{2}} - \frac{b ^{6}}{y ^{2}} = (a^{2} + b^{2})^{2}$ (3) $\frac{a ^{6}}{x ^{2}} - \frac{b ^{6}}{y ^{2}} = (a^{2} - b^{2})^{2}$ (4) $\frac{a ^{6}}{x ^{2}} + \frac{b ^{6}}{y ^{2}} = (a^{2} - b^{2})^{2}$ 187. $Let P (3, 3) be a point on the hyperbola, \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 . If the normal to it at P intersects the x -axis at (9, 0) and e is its eccentricity, then the ordered pair (a^{2}, e^{2}) is equal to:$ (1) $(9, 3)$ (2) $(\frac{9}{2}, 3)$ (3) $(\frac{3}{2}, 2)$ (4) $(\frac{9}{2}, 2)$ 188. $A hyperbola passes through the foci of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 and its transverse and conjugate axes coincidewith major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then theequation of the hyperbola is:$ (1) $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ (2) $\frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1$ (3) $x^{2} - y^{2} = 9$ (4) $\frac{x ^{2}}{9} - \frac{y ^{2}}{25} = 1$ 189. $The pole of the straight line 9 x + y - 28 = 0 with respect to the circle 2 x^{2} + 2 y^{2} - 3 x + 5 y - 7 = 0 is$ (1) $(3, 1)$ (2) $(3, - 1)$ (3) $(- 3, 1)$ (4) $(4, - 8)$ 190. $If y = mx + 4 (m > 0) is a tangent to the hyperbola \frac{x ^{2}}{25} - \frac{y ^{2}}{9} = 1, then the point of contact of this tangent is$ (1) $(- \frac{25}{4}, - \frac{9}{4})$ (2) $(\frac{25}{4}, \frac{9}{4})$ (3) $(1, 5)$ (4) $(- \frac{1}{2}, \frac{7}{2})$ 191. $Find the equation of circle having normals (x - 1) (y - 2) = 0 and a tangent 3 x + 4 y = 6 ?$ (1) $(x - 1)^{2} + (y - 2)^{2} = 1$ (2) $(x - 2)^{2} + (y - 1)^{2} = 1$ (3) $(x + 1)^{2} + (y + 2)^{2} = 1$ (4) $(x + 2)^{2} + (y + 1)^{2} = 1$ 192. $From a point (C, 0) three normals are drawn to the parabola y^{2} = x . Then,$ (1) $C < \frac{1}{2}$ (2) $C = \frac{1}{2}$ (3) $C > \frac{1}{2}$ (4) $\frac{1}{2} > C > \frac{1}{4}$ 193. $To which point the origin is to be shifted in order to eliminate first powers of x and y (x^{1}) and y^{1} terms) from the equation 4 x^{2} + 9 y^{2} - 8 x + 36 y + 4 = 0 ?$ (1) $(1, 2)$ (2) $(- 1, 2)$ (3) $(1, - 2)$ (4) $(- 1, - 3)$ 194. $If the circles x^{2} + y^{2} - 2 λx - 2 y - 7 = 0 and 3 (x^{2} + y^{2}) - 8 x + 29 y = 0 are orthogonal, then λ is equal to$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 195. $If a chord, which is not a tangent, of the parabola y^{2} = 16x has the equation 2x + y = p, and midpoint (h, k), then which of the following is(are) possible value(s) of p, h and k?$ (1) $p = -1, h = 1, k = -3$ (2) $p = 2, h = 3, k = -4$ (3) $p = -2, h = 2, k = -4$ (4) $p = 5, h = 4, k = -3$ 196. $Two chords of the circle x^{2} + y^{2} - 2 gx - 2 hy + g^{2} + h^{2} - c^{2} = 0 are passing through the point (g, h + c) and the line y = x bisects these two chords. Then$ (1) $4 g^{2} - 4 h^{2} - 8 gh + 4 hc - 4 gc - c^{2} = 0$ (2) $4 g^{2} + 4 h^{2} - 8 gh + 4 hc - 4 gc - c^{2} < 0$ (3) $4 g^{2} + 4 h^{2} + 8 gh + 4 hc + 4 gc + c^{2} = 0$ (4) $4 g^{2} + 4 h^{2} - 8 gh + 4 hc - 4 gc - c^{2} > 0$ 197. $Let P be the point on the parabola, y^{2} = 8x which is at a minimum distance from the centre C of the circle, x^{2} + (y + 6)^{2} = 1. Then the equation of the circle, passing through C and having its centre at P is :$ (1) $x^{2} + y^{2} - x + 4y -12 = 0$ (2) $x^{2} + y^{2} - \frac{x}{4} + 2y -24 = 0$ (3) $x^{2} + y^{2} -4 x + 9y + 18 = 0$ (4) $x^{2} + y^{2} - 4x + 8y +12 = 0$ 198. $The equation of the circle passing through (0, 0) and which makes intercepts a and b on the coordinates axes is$ (1) $x^{2} + y^{2} + ax + by = 0$ (2) $x^{2} + y^{2} + ax - by = 0$ (3) $x^{2} + y^{2} - ax + by = 0$ (4) $x^{2} + y^{2} - ax - by = 0$ 199. $The length of the common chord of the two circles (x - a)^{2} + y^{2} = a^{2} and x^{2} + (y - b)^{2} = b^{2}, is$ (1) $\frac{ab}{a ^{2} + b ^{2}}$ (2) $\frac{2 ab}{a ^{2} + b ^{2}}$ (3) $\frac{a + b}{a ^{2} + b ^{2}}$ (4) $a^{2} + b^{2}$ 200. $The normal drawn at P (- 1, 2) on the circle x^{2} + y^{2} - 2 x - 2 y - 3 = 0 meets the circle at another point Q . Then, the coordinates of Q are$ (1) $(3, 0)$ (2) $(- 3, 0)$ (3) $(2, 0)$ (4) $(- 2, 0)$ 201. $Let T and C respectively be the transverse and conjugate axes of the hyperbola 16 x^{2} - y^{2} + 64 x + 4 y + 44 - 0 . Then the area of the region above the parabola x^{2} = y + 4, below the transverse axis T and on the right of the conjugate axis C is$ (1) $46 + \frac{44}{3}$ (2) $46 + \frac{28}{3}$ (3) $46 - \frac{44}{3}$ (4) $46 - \frac{28}{3}$ 202. $The locus of the centroid of the triangle formed by any point P on the hyperbola 16 x^{2} - 9 y^{2} + 32x + 36y - 164 = 0, and its foci is :$ (1) $16 x^{2} - 9 y^{2} + 32x + 36y - 36 = 0$ (2) $9 x^{2} - 16 y^{2} + 36x + 32y - 144 = 0$ (3) $16 x^{2} - 9 y^{2} + 32x + 36y - 144 = 0$ (4) $9 x^{2} - 16 y^{2} + 36x + 32y - 36 = 0$ 203. $Let λ x - 2 y = μ be a tangent to the hyperbola a^{2} x^{2} - y^{2} = b^{2} . Then (\frac{λ}{a})^{2} - (\frac{μ}{b})^{2} is equal to:$ (1) $-2$ (2) $-4$ (3) $2$ (4) $4$ 204. $The equation of the transverse common tangent of the circles x^{2} + y^{2} - 6 x - 8 y + 9 = 0 and x^{2} + y^{2} + 2 x - 2 y + 1 = 0 is$ (1) $4 x + 3 y - 4 = 0$ (2) $3 x + y - 1 = 0$ (3) $2 x - y + 2 = 0$ (4) $x + 2 y - 3 = 0$ 205. $Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2 a and 2 b, respectively, and one focus and the corresponding directrix of this hyperbola be (- 5, 0) and 5 x + 9 = 0, respectively. If the product of the focal distances of a point (α, 25) on the hyperbola is p, then 4 p is equal to _________.$ 206. $Let the foci and length of the latus rectum of an ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b be (\pm 5, 0) and 50, respectively. Then, the square of the eccentricity of the hyperbola \frac{x ^{2}}{b ^{2}} - \frac{y ^{2}}{a ^{2} b ^{2}} = 1 equals$ 207. $If the equation of the hyperbola with foci (4, 2) and (8, 2) is 3 x^{2} - y^{2} - αx + βy + γ = 0, then α + β + γ is equal to ________.$ 208. $If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x - 3)2 + (y + 2)2 = r2, then the value of r2 is$ 209. $Let m_{1} and m_{2} be the slopes of the tangents drawn from the point P(4,1) to the hyperbola H = \frac{y ^{2}}{25} - \frac{x ^{2}}{16} = 1 If Q is the point from which the tangents drawn to H have slopes \| m_{1} \| and \| m_{2} \| and they make positive intercepts α and β on the x-axis, then \frac{( P Q ) ^{2}}{α β} is equal to________.$ 210. $Let S be the focus of the hyperbola \frac{x ^{2}}{3} - \frac{y ^{2}}{5} = 1, on the positive x -axis. Let C be the circle with its centre at A (6, 5) and passing through the point S . If O is the origin and SAB is a diameter of C, then the square of the area of the triangle OSB is equal to________$