Hyperbola PYQs Set 9 (PDF)

Set 9 of 11 · Questions 241-270 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 9 (Q241-270)
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241. $Let C be the circle centre (0, 0) and radius 3 units. The equation of the locus of the mid-points of the chords of the circle c that subtends an angle of \frac{2 π}{3} at its centre is$ (1) $x^{2} + y^{2} = \frac{1}{4}$ (2) $x^{2} + y^{2} = \frac{27}{4}$ (3) $x^{2} + y^{2} = \frac{9}{4}$ (4) $x^{2} + y^{2} = \frac{5}{4}$ 242. $If the circle x^{2} + y^{2} - 6 x + 2 y = 28 cuts off a chord of length λ units of the line 2 x - 5 y + 18 = 0, then the value of λ is$ (1) $3$ (2) $6$ (3) $12$ (4) $9$ 243. $If the common tangent to the parabolas, y^{2} = 4 x and x^{2} = 4 y also touches the circle, x^{2} + y^{2} = c^{2}, then c is equal to:$ (1) $\frac{1}{2}$ (2) $\frac{1}{4}$ (3) $\frac{1}{2 2}$ (4) $\frac{1}{2}$ 244. $Let the foci of a hyperbola H coincide with the foci of the ellipse E : \frac{( x - 1 ) ^{2}}{100} + \frac{( y - 1 ) ^{2}}{75} = 1 and the eccentricity of the hyperbola H be the reciprocal of the eccentricity of the ellipse E . If the length of the transverse axis of H is α and the length of its conjugate axis is β, then 3 α^{2} + 2 β^{2} is equal to$ (1) $237$ (2) $242$ (3) $205$ (4) $225$ 245. $Consider the hyperbola H : x2 - y2 = 1 and a circle S with center N(x2, 0). Suppose that H and S touch each other at a point P(x1, y1) with x1 > 1 and y1 > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle ΔPMN, then the correct expression(s) is(are)$ (1) $\frac{d l}{d x _{1}} = 1 - \frac{1}{3 x _{1}^{2}} for x1 > 1$ (2) $\frac{d m}{d x _{1}} = \frac{x _{1}}{3 x _{1}^{2} - 1} for x_{1} > 1$ (3) $\frac{d l}{d x _{1}} = 1+ \frac{1}{3 x _{1}^{2}} for x1 > 1$ (4) $\frac{d m}{d y _{1}} = \frac{1}{3} for y1 > 0$ 246. $The locus of the mid points of the chords of the hyperbola x^{2} - y^{2} = 4, which touch the parabola y^{2} = 8x, is :$ (1) $y^{3} (x - 2) = x^{2}$ (2) $x^{3} (x - 2) = y^{2}$ (3) $y^{2} (x - 2) = x^{3}$ (4) $x^{2} (x - 2) = y^{3}$ 247. $If a circle with radius 2.5 units passes through the points (2, 3) and (5, 7), then its centre is$ (1) $(1.5, 2)$ (2) $(7, 10)$ (3) $(3, 4)$ (4) $(3.5, 5)$ 248. $O (0, 0) and A (1, 0) are centres of two unit circles C_{1} and C_{2} respectively. C_{3} is also a unit circle having its centre above X -axis and passing through O and A . The equation of the common tangent to C_{1} and C_{3} which does not intersect the circle C_{2} is$ (1) $3 x - y + 2 = 0$ (2) $x + 3 y + 2 = 0$ (3) $3 x - y - 2 = 0$ (4) $x + 3 y - 2 = 0$ 249. $If the transverse and conjugate axes of hyperbola arc equal, then its eccentricity is$ (1) $3$ (2) $2$ (3) $\frac{1}{2}$ (4) $2$ 250. $If one of the diameter of the circle x^{2} + y^{2} - 2 x - 6 y + 6 = 0 is a chord to the circle with centre (2, 1), then the radius of the bigger circle is$ (1) $6$ (2) $4$ (3) $2$ (4) $3$ 251. $The equation of the circle having x - y - 2 = 0 and x - y + 2 = 0 as two tangents and x + y = 0 as a diameter is$ (1) $x^{2} + y^{2} + 2 x - 2 y + 1 = 0$ (2) $x^{2} + y^{2} - 2 x + 2 y - 1 = 0$ (3) $x^{2} + y^{2} = 2$ (4) $x^{2} + y^{2} = 1$ 252. $Let the sum of the focal distances of the point P (4, 3) on the hyperbola H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 be 8 \frac{5}{3} . If for H, the length of the latus rectum is l and the product of the focal distances of the point P is m, then 9 l^{2} + 6 m is equal to:$ (1) $187$ (2) $184$ (3) $186$ (4) $185$ 253. $If a directrixof a hyperbola centred at the origin and passing through the point (4, - 23) is 5 x = 45 and its eccentricity is e, then:$ (1) $4 e^{4} - 24 e^{2} + 27 = 0$ (2) $4 e^{4} - 12 e^{2} - 27 = 0$ (3) $4 e^{4} - 24 e^{2} + 35 = 0$ (4) $4 e^{4} + 8 e^{2} - 35 = 0$ 254. $From a point A (0, 3) on the circle (x + 2)^{2} + (y - 3)^{2} = 4, a chord AB is drawn and it is extended to a point Q such that AQ = 2 AB . Then the locus of Q is$ (1) $(x + 4)^{2} + (y - 3)^{2} = 16$ (2) $(x + 1)^{2} + (y - 3)^{2} = 32$ (3) $(x + 1)^{2} + (y - 3)^{2} = 4$ (4) $(x + 1)^{2} + (y - 3)^{2} = 1$ 255. $The angle between the pair of tangents drawn from (1, 1) to the circle x^{2} + y^{2} + 4 x + 4 y - 1 = 0 is$ (1) $\frac{π}{2}$ (2) $\frac{π}{4}$ (3) $\frac{π}{3}$ (4) $\frac{π}{6}$ 256. $Let P be the point of intersection of the common tangents to the parabola y^{2} = 12 x and hyperbola 8 x^{2} - y^{2} = 8 . If S and S^{’} denote the foci of the hyperbola where S lies on the positive x -axis then P divides SS^{’} in a ratio:$ (1) $13 : 11$ (2) $14 : 13$ (3) $5 : 4$ (4) $2 : 1$ 257. $The length of common chord of the circles x^{2} + y^{2} - 6 x - 4 y + 13 - c^{2} = 0 and x^{2} + y^{2} - 4 x - 6 y + 13 - c^{2} = 0 is$ (1) $4 c^{2} - 2$ (2) $\frac{1}{2} 4 c^{2} - 2$ (3) $c^{2} - 2$ (4) $4 c^{2} - 1$ 258. $If A, B are the points of contact of the tangents drawn from the point P (- 2, - 3) to the circle x^{2} + y^{2} - 8 x - 10 y + 5 = 0 and the chord AB subtends an angle θ at P then tan θ =$ (1) $\frac{3}{4}$ (2) $\frac{24}{7}$ (3) $\frac{7}{24}$ (4) $\frac{4}{3}$ 259. $The sum of the squares of the distances of a moving point from 2 fixed points A (a, 0) and B (- a, 0) is equal to a constant 2 c^{2}, then the equation of its locus is$ (1) $x^{2} + y^{2} = c^{2} - a^{2}$ (2) $x^{2} + y^{2} = c^{2} + a^{2}$ (3) $2 x^{2} + 2 y^{2} = c^{2} + a^{2}$ (4) $2 x^{2} - 2 y^{2} = c^{2} + a^{2}$ 260. $If a circle C passing through (4, 0) touches the circle x^{2} + y^{2} + 4 x - 6 y - 12 = 0 externally at the point (1, - 1), then the radius of C is$ (1) $12$ (2) $4$ (3) $3$ (4) $5$ 261. $If the line x-1 = 0, is a directrix of the hyperbola k x^{2} - y^{2} =6, then the hyperbola passes through the point$ (1) $(- 25, 6)$ (2) $(- 5, 3)$ (3) $(5, - 2)$ (4) $(25, 36)$ 262. $Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 . Let e' and l' respectively the eccentricity and length of the latus rectum of its conjugate hyperbola If e^{2} = \frac{11}{14} l and (e^{'})^{2} = \frac{11}{8} l^{'}, then the value of 77a+ 44b is equal to$ (1) $100$ (2) $110$ (3) $120$ (4) $130$ 263. $A circle has its centre in the first quadrant and passes through (2, 3) . If this circle makes intercepts of length 3 and 4 respectively on x = 2 and y = 3, its equation is$ (1) $x^{2} + y^{2} + 3 x - 5 y + 8 = 0$ (2) $x^{2} + y^{2} - 4 x - 6 y + 13 = 0$ (3) $x^{2} + y^{2} - 6 x - 8 y + 23 = 0$ (4) $x^{2} + y^{2} - 8 x - 9 y + 30 = 0$ 264. $If A (\frac{π}{3}), B (\frac{π}{6}) are the points on the circle represented in parametric from with centre (0, 0) and radius 12 then the length of the chord AB is$ (1) $6 (6 - 2)$ (2) $6 (6 - 3)$ (3) $2 (3 - 1)$ (4) $6 (3 - 1)$ 265. $The equation of the incircle of the triangle formed by the lines x = 0, y = 0 and 3 x + 4 y - 24 = 0 is$ (1) $x^{2} + y^{2} - 24 x - 24 y + 144 = 0$ (2) $x^{2} + y^{2} - 6 x - 6 y + 9 = 0 .$ (3) $x^{2} + y^{2} - 4 x - 4 y + 4 = 0$ (4) $x^{2} + y^{2} - 8 x - 8 y + 16 = 0$ 266. $An ellipse E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 passes through the vertices of the hyperbola H : \frac{x ^{2}}{49} - \frac{y ^{2}}{64} = - 1 . Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H. Let the product of the eccentricities of E and H be \frac{1}{2} . If l is the length of the latus rectum of the ellipse E, then the value of 113 l is equal to-----------.$ 267. $Let the circle C touch the line x - y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length \frac{4}{13} along the line - 3 x + 2 y = 1 . Let H be the hyperbola \frac{x ^{2}}{α ^{2}} - \frac{y ^{2}}{β ^{2}} = 1, whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C . Then 2 α^{2} + 3 β^{2} is equal to .$ 268. $Let the tangent to the parabola y^{2} = 12x at the point (3, α) be perpendicular to the line 2x + 2y = 3. Then the square of distance of the point (6, - 4) from the normal to the hyperbola α^{2} x^{2} - 9 y^{2} = 9 α^{2} at its point (α -1, α + 2) is equal to.............$ 269. $Let a line L_{1} be tangent to the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{4} = 1 and let L_{2} be the line passing through the origin and perpendicular to L_{1} If the locus of the point of intersection of L_{1} and L_{2} is (x^{2} + y^{2})^{2} = α x^{2} + β y^{2}, then α + β is equal to______.$ 270. $For the hyperbola H : x^{2} - y^{2} = 1 and the ellipse E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b > 0, let the (1) eccentricity of E be reciprocal of the eccentricity of H, and (2) the line y = \frac{5}{2} x + K be a common tangent of E and H. Then 4(a^{2} + b^{2}) is equal to_______.$