Hyperbola PYQs Set 8 (PDF)

Set 8 of 11 · Questions 211-240 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 8 (Q211-240)
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211. $Let the foci of a hyperbola be (1, 14) and (1, - 12) . If it passes through the point (1, 6), then the length of its latus-rectum is:$ (1) $\frac{24}{5}$ (2) $\frac{25}{6}$ (3) $\frac{144}{5}$ (4) $\frac{288}{5}$ 212. $The equation of a circle passing through the point (2, 8), touching the lines 4 x - 3 y - 24 = 0 and 4 x + 3 y - 42 = 0 and having the x coordinate of its centre less than or equal to 8 is$ (1) $x^{2} + y^{2} + 2 x - 8 y - 8 = 0$ (2) $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ (3) $x^{2} + y^{2} + 4 x - 10 y + 4 = 0$ (4) $x^{2} + y^{2} - 6 x - 4 y - 24 = 0$ 213. $Let e_{1} be the eccentricity of the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1 and e_{2} be the eccentricity of the ellipse \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b, which passes through the foci of the hyperbola. If e_{1} e_{2} = 1, then the length of the chord of the ellipse parallel to the x -axis and passing through (0, 2) is :$ (1) $45$ (2) $\frac{8 5}{3}$ (3) $\frac{10 5}{3}$ (4) $35$ 214. $Let A be the centre of the circle x^{2} + y^{2} - 2 x - 4 y - 20 = 0 . If the tangents drawn at the points B (1, 7) and D (4, - 2) on the given circle meet at the point C, then the area of the quadrilateral ABCD is$ (1) $60$ (2) $65$ (3) $70$ (4) $75$ 215. $A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x -axis. Then the eccentricity of the hyperbola is:$ (1) $\frac{3}{2}$ (2) $3$ (3) $2$ (4) $\frac{2}{3}$ 216. $The normal to the circle given by x^{2} + y^{2} - 6 x + 8 y - 144 = 0 at (8, 8) meets the circle again at the point$ (1) $(2, - 16)$ (2) $(2, 16)$ (3) $(- 2, 16)$ (4) $(- 2, - 16)$ 217. $For the hyperbola \frac{x ^{2}}{co s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1, which of the following remains constant when α varies ?$ (1) $Eccentricity$ (2) $Directrix$ (3) $Abscissae of vertices$ (4) $Abscissae of foci$ 218. $The radius of a circle whose centre is (2, 1) and one of the chords is a diameter of the circle x^{2} + y^{2} - 2 x - 6 y + 6 = 0, is ............ units.$ (1) $3$ (2) $4$ (3) $2$ (4) $1$ 219. $The combined equation of the direct common tangents of the circles x^{2} + y^{2} + 2 x = 0 and x^{2} + y^{2} - 2 y - 3 = 0 is$ (1) $xy + x + 2 y + 2 = 0$ (2) $x^{2} - xy - 2 y^{2} + 3 x - 6 y = 0$ (3) $2 x^{2} + 5 xy + 2 y^{2} + 13 x + 14 y + 20 = 0$ (4) $2 x^{2} - 9 xy + 9 y^{2} + 3 x - 6 y + 1 = 0$ 220. $The locus of the midpoints of the chord of the circle, x^{2} + y^{2} = 25 which is tangent to the hyperbola, \frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1 ¯ is:$ (1) $(x^{2} + y^{2})^{2} - 16 x^{2} + 9 y^{2} = 0$ (2) $(x^{2} + y^{2})^{2} - 9 x^{2} + 144 y^{2} = 0$ (3) $(x^{2} + y^{2})^{2} - 9 x^{2} - 16 y^{2} = 0$ (4) $(x^{2} + y^{2})^{2} - 9 x^{2} + 16 y^{2} = 0$ 221. $Consider the hyperbola H : x2 - y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle ΔPMN, then the correct expression(s) is(are)$ (1) $\frac{d l}{d x _{1}} = 1 - \frac{1}{3 x _{1}^{2}} for x1 > 1$ (2) $for x1 > 1$ (3) $\frac{d l}{d x _{1}} = 1+ \frac{1}{3 x _{1}^{2}} for x1 > 1$ (4) $\frac{d m}{d y _{1}} = \frac{1}{3} for y1 > 0$ 222. $Two straight rods of lengths 2 a and 2 b move along the coordinate axes in such a way that their extremities are always concyclic. Then the locus of the centres of such circles is$ (1) $2 x^{2} + y^{2} = a^{2} + b^{2}$ (2) $2 (x^{2} - y^{2}) = a^{2} + b^{2}$ (3) $x^{2} + y^{2} = a^{2} + b^{2}$ (4) $x^{2} - y^{2} = a^{2} - b^{2}$ 223. $Consider a hyperbola H : x^{2} - 2 y^{2} = 4 . Let the tangent at a point P (4, 6) meet the x - axis at Q and latusrectum at R (x_{1}, y_{1}), x_{1} > 0 . If F is a focus of H which is nearer to the point P, then the area of Δ QFR isequal to$ (1) $46$ (2) $6 - 1$ (3) $\frac{7}{6} - 2$ (4) $46 - 1$ 224. $The length of the common chord of the two circles x^{2} + y^{2} - 4 x - 8 y + 4 = 0 and x^{2} + y^{2} - 8 x - 12 y + 16 = 0 is$ (1) $46$ (2) $15$ (3) $55$ (4) $3$ 225. $Equation of the circle centered at (4, 3) touching the circle x^{2} + y^{2} = 1 externally, is$ (1) $x^{2} + y^{2} + 8 x + 6 y + 9 = 0$ (2) $x^{2} + y^{2} - 8 x - 6 y + 9 = 0$ (3) $x^{2} + y^{2} - 8 x + 6 y + 9 = 0$ (4) $x^{2} + y^{2} + 8 x - 6 y + 9 = 0$ 226. $If P (θ) = (x_{1}, \frac{3 5}{2}), 0 < θ < \frac{π}{2} is a point on the hyperbola \frac{x ^{2}}{25} - \frac{y ^{2}}{9} = 1, where θ is the parameter in its parametric form, then 2 x_{1} + 9 sin^{2} θ =$ (1) $8$ (2) $10$ (3) $20$ (4) $34$ 227. $The line ax + by + c = 0 is normal to the circle x^{2} + y^{2} + 2 gx + 2 fy + d = 0 if$ (1) $ag + bf + c = 0$ (2) $ag + bf - c = 0$ (3) $ag - bf + c = 0$ (4) $ag - bf - c = 0$ 228. $A circle having centre at the origin passes through the three vertices of an equilateral triangle the length of its median being 9 units. Then the equation of that circle is$ (1) $x^{2} + y^{2} = 9$ (2) $x^{2} + y^{2} = 18$ (3) $x^{2} + y^{2} = 36$ (4) $x^{2} + y^{2} = 81$ 229. $If a circle C_{1} : x^{2} + y^{2} = 16 intersects another circle C_{2} with radius 5 such that the common chord is of maximum length and has a slope equal to \frac{3}{4} then the centre of the circle C_{2} is$ (1) $(- \frac{9}{5}, \frac{12}{5})$ (2) $(\frac{9}{5}, \frac{12}{5})$ (3) $(- \frac{5}{9}, \frac{6}{5})$ (4) $(\frac{7}{5}, \frac{12}{5})$ 230. $The locus of the mid points of the chords of the circle x^{2} + y^{2} = 4 which subtend a right angle at the origin is$ (1) $x + y = 1$ (2) $x + y = 2$ (3) $x^{2} + y^{2} = 1$ (4) $x^{2} + y^{2} = 2$ 231. $The equation of two circles which touch the Y -axis at (0, 3) and make an intercept of 8 units on X -axis are$ (1) $x^{2} + y^{2} \pm 10 x - 6 y + 9 = 0$ (2) $x^{2} + y^{2} \pm 6 x - 10 y + 9 = 0$ (3) $x^{2} + y^{2} - 8 x \pm 10 y + 9 = 0$ (4) $x^{2} + y^{2} + 10 x \pm 6 y + 9 = 0$ 232. $The length of the diameter of the circle which cuts three circles x^{2} + y^{2} - x - y - 14 = 0; x^{2} + y^{2} + 3 x - 5 y - 10 = 0; x^{2} + y^{2} - 2 x + 3 y - 27 = 0 orthogonally, is$ (1) $4$ (2) $2$ (3) $8$ (4) $6$ 233. $A circle S = 0 with radius 2 touches the line x + y - 2 = 0 at (1, 1) . Then, the length of the tangent drawn from the point (1, 2) to S = 0 is$ (1) $1$ (2) $2$ (3) $3$ (4) $2$ 234. $If the axes are rotated anticlockwise through an angle 90^{\circ}, then the equation x^{2} = 4 ay is changed to the equation$ (1) $y^{2} = 4 ax$ (2) $x^{2} = - 4 ay$ (3) $y^{2} = - 4 ax$ (4) $x^{2} = 4 ay$ 235. $The locus of the vertices of the family of parabolas y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2a is$ (1) $xy = \frac{3}{4}$ (2) $xy = \frac{35}{16}$ (3) $xy = \frac{64}{105}$ (4) $xy = \frac{105}{64}$ 236. $The straight line 2 x + 3 y - k = 0, k > 0 cuts the X and Y \times axes at A and B . The area of △ OAB, where O is the origin, is 12 sq. units. The equation of the circle having AB as diameter is$ (1) $x^{2} + y^{2} + 4 x - 6 y = 0$ (2) $x^{2} + y^{2} - 6 x - 4 y = 0$ (3) $x^{2} + y^{2} - 4 x - 6 y = 0$ (4) $x^{2} + y^{2} - 6 x + 4 y = 0$ 237. $Consider the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 having one of its focus at P (- 3, 0) . If the latus ractum through its other focus subtends a right angle at P and a^{2} b^{2} = α 2 - β, α, β \in N, then α + β is________.$ 238. $A square ABCD has all its vertices on the curve x^{2} y^{2} = 1 . The midpoints of its sides also lie on the samecurve. Then, the square of area of ABCD is$ 239. $Let H_{n} : \frac{x ^{2}}{1 + n} - \frac{y ^{2}}{3 + n} = 1, n \in N . Let k be the smallest even value of n such that the eccentricity of H_{k} is a rational number. If l is the length of the latus rectum of H_{k}, then 21 l is equal to$ 240. $The line 2x + y = 1 is tangent to the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 . If this line passes through the point of intersection of the nearest directrix and the x-axis, then find the eccentricity of the hyperbola$