Hyperbola PYQs Set 10 (PDF)

Set 10 of 11 · Questions 271-300 of 302

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 10 (Q271-300)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
271. $The area (in sq. units) of the smaller of the two circles that touch the parabola, y^{2} = 4 x at the point (1, 2) and the x -axis is:$ (1) $8 π (2 - 2)$ (2) $4 π (2 - 2)$ (3) $4 π (3 + 2)$ (4) $8 π (3 - 22$ 272. $The chord of the circle x^{2} + y^{2} - 4 x = 0 which is bisected at (1, 0) is perpendicular to the line$ (1) $x + y = 0$ (2) $y = x$ (3) $y = 1$ (4) $x = 1$ 273. $The equation of a tangent to the parabola, x^{2} = 8 y which makes an angle θ with the positive direction of x -axis, is:$ (1) $y = x tan θ + 2 cot θ$ (2) $y = x tan θ - 2 cot θ$ (3) $x = y cot θ + 2 tan θ$ (4) $x = y cot θ - 2 tan θ$ 274. $Let A = (0, 4) and B = (2 cos θ, 2 sin θ), for some 0 < θ < \frac{π}{2} . Let P divide the line segment AB in the ratio 2 : 3 internally. The locus of P is$ (1) $Circle$ (2) $Ellipse$ (3) $Parabola$ (4) $Hyperbola$ 275. $The angle between the two tangents drawn from origin to the circle x^{2} + y^{2} - 14 x + 2 y + 25 = 0 is$ (1) $0^{\circ}$ (2) $45^{\circ}$ (3) $90^{\circ}$ (4) $60^{\circ}$ 276. $If the angle between a pair of tangents drawn from a point P to the circle x^{2} + y^{2} + 4 x - 6 y + 9 sin^{2} α + 13 cos^{2} α = 0 is 2 α, then the equation of the locus of P is$ (1) $x^{2} + y^{2} + 4 x - 6 y + 4 = 0$ (2) $x^{2} + y^{2} + 4 x - 6 y - 9 = 0$ (3) $x^{2} + y^{2} - 4 x + 6 y - 4 = 0$ (4) $x^{2} + y^{2} + 4 x - 6 y + 9 = 0$ 277. $Let S be the focus of the hyperbola x^{2} - 2 y^{2} = 1 lying on the positive X -axis. Let P (- 1, 1) be a given point. Then the area of the triangle formed by the line PS with the coordinate axes is (in sq. units)$ (1) $\frac{2}{2 ( 2 + 3 )}$ (2) $\frac{6}{2 ( 2 + 6 )}$ (3) $\frac{3}{2 ( 2 + 6 )}$ (4) $\frac{3}{2 ( 2 + 3 )}$ 278. $If the latus rectum of a hyperbola subtends an angle of 120^{\circ} at its centre, then its eccentricity is$ (1) $\frac{3 + 2}{2}$ (2) $\frac{3 + 5}{2}$ (3) $\frac{3 - 2}{3}$ (4) $\frac{3 + 7}{2}$ 279. $The area of an equilateral triangle inscribed in the circle x^{2} + y^{2} - 6 x + 2 y - 28 = 0 is sq. units$ (1) $\frac{27 3}{2}$ (2) $\frac{37 3}{2}$ (3) $\frac{31 3}{2}$ (4) $\frac{57 3}{2}$ 280. $Let S be the focus of the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1 lying on the positive X -axis and P (5, y_{1}) be point on the hyperbola. Then SP =$ (1) $\frac{1}{4}$ (2) $\frac{3}{4}$ (3) $\frac{9}{4}$ (4) $\frac{5}{4}$ 281. $The area (in sq units) of the triangle formed by the tangent, normal at (1, 3) to the circle x^{2} + y^{2} = 4 and the X -axis, is$ (1) $43$ (2) $\frac{7}{2} 3$ (3) $23$ (4) $\frac{1}{2} 3$ 282. $The straight line x + 2 y = 1 cuts the X -axis at A and Y -axis at B . A circle is drawn through A, B and the origin. The sum of the perpendicular distances from A, B on to the tangent drawn at origin to the circle S is$ (1) $equal to the radius of the circle S$ (2) $equal to the diameter of the circle S$ (3) $equal to twice the diameter of the circle S$ (4) $equal to 5 times the radius of the circle S$ 283. $The point of intersection of the common tangents drawn to the circles x^{2} + y^{2} - 4 x - 2 y + 1 = 0 and x^{2} + y^{2} - 6 x - 4 y + 4 = 0, is$ (1) $(\frac{5}{2}, \frac{3}{2})$ (2) $(\frac{6}{5}, \frac{1}{5})$ (3) $(0, - 1)$ (4) $(\frac{12}{5}, \frac{7}{5})$ 284. $The equation of the line which is tangent to both the circle x^{2} + y^{2} = 5 and the parabola y^{2} = 40 x is$ (1) $2 x - y \pm 5 = 0$ (2) $2 x - y + 5 = 0$ (3) $2 x - y - 5 = 0$ (4) $2 x + y + 5 = 0 .$ 285. $If l x + m y = 1 is a normal to the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, then a^{2} m^{2} - b^{2} l^{2} =$ (1) $\frac{m ^{2}}{l ^{2}} (a^{2} + b^{2})^{2}$ (2) $(l^{2} + m^{2}) (a^{2} + b^{2})^{2}$ (3) $\frac{l ^{2}}{m ^{2}} (a^{2} + b^{2})^{2}$ (4) $l^{2} m^{2} (a^{2} + b^{2})^{2}$ 286. $For any value of θ, if the straight lines x sin θ + (1 - cos θ) y = a sin θ and x sin θ - (1 + cos θ) y + a sin θ = 0 intersect at P (θ), then the locus of P (θ) is a$ (1) $straight line$ (2) $circle$ (3) $parabola$ (4) $hyperbola$ 287. $The asymptotes of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, with any tangent to the hyperbola form a triangle whose area is a^{2} tan (α) . Then, its eccentricity equals$ (1) $sec (α)$ (2) $cosec α$ 288. $If S \equiv \frac{x ^{2}}{k - 7} + \frac{y ^{2}}{11 - k} - 1 = 0, k \in R - {7, 11}, then which one of the following statements is incorrect?$ (1) $S = 0 represents a circle with radius 2, when k = 9$ (2) $S = 0 represents an ellipse with eccentricity \frac{2}{3}, when k = 10$ (3) $S = 0 represents a hyperbola with eccentricity \frac{6}{5}, when k = 12$ (4) $S = 0 represents a hyperbola with eccentricity \frac{3}{2}, when k = 13$ 289. $If the line 2 x + 6 y = 2 touches the hyperbola x^{2} - 2 y^{2} = 4, then the coordinates of the point of contact are$ (1) $(\frac{1}{2}, \frac{1}{6})$ (2) $(4, - 6)$ (3) $(4, 6)$ (4) $(- 2, 6)$ 290. $Assertion : The distance between the points p (\frac{π}{4}) and p (\frac{π}{3}) on the hyperbola 9 x^{2} + 16 y^{2} = 9 is \frac{1}{2 2} 66 - 332 - 93 Reason : x = a cosh t, y = b sinh t are the parametric equations of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 The correct option among the following is$ (1) $(A) is true, (R) is true and (R) is the correct explanation for (A)$ (2) $(A) is true, (R) is true but (R) is not the correct explanation for (A)$ (3) $(A) is true but (R) is false$ (4) $(A) is false but (R) is true$ 291. $Let a tangent to the curve y^{2} = 24 x meet the curve x y = 2 at the points A and B. Then the mid points of such line segments AB lie on a parabola with the$ (1) $directrix 4x=3$ (2) $directric 4x=-3$ (3) $Length of latus rectum \frac{3}{2}$ (4) $Length of latus rectum 2$ 292. $If a hyperbola passes through the focii of the ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1. Its transverse and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1, then$ (1) $the equation of hyperbola is \frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1$ (2) $the equation of hyperbola is \frac{x ^{2}}{9} - \frac{y ^{2}}{25} = 1$ (3) $focus of hyperbola is (5, 0)$ (4) $focus of hyperbola is (-5, 0)$ 293. $Let a line L : 2 x + y = k, k > 0 be a tangent to the hyperbola x^{2} - y^{2} = 3 . If L is also a tangent to the parabola y^{2} = α x, then α is equal to:$ (1) $12$ (2) $-12$ (3) $24$ (4) $-24$ 294. $If the circle S \equiv x^{2} + y^{2} - 4 = 0 intersects another circle S^{’} = 0 of radius \frac{5 2}{2} in such a manner that the common chord is of maximum length with slope equal to \frac{1}{4}, then the centre of S^{’} = 0 is$ (1) $(- 1, 4) or (1, - 4)$ (2) $(- \frac{2}{2}, 22) or (\frac{2}{2}, - 22)$ (3) $(- 22, \frac{2}{2}) or (22, \frac{- 2}{2})$ (4) $(4, - 1) or (- 4, 1)$ 295. $Let the equation of two diameters of a circle x^{2} + y^{2} -2x + 2fy + 1 = 0 be 2px - y = 1 and 2x + py = 4p. Then the slope m \in_{(} 0, \infty_{)} of the tangent to the hyperbola 3 x^{2} - y^{2} = 3 passing through the centre of the circle is equal to_______.$ 296. $A common tangent T to the curves C_{1} : \frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1 and C_{2} : \frac{x ^{2}}{42} - \frac{y ^{2}}{143} = 1 does not pass through the fourth quadrant. If T touches C_{1} at (x_{1}, y_{1}) and C_{2} at (x_{2}, y_{2}), then \| 2 x_{1} + x_{2} \| is equal to$ 297. $Let the latus rectum of the hyperbola \frac{x ^{2}}{9} - \frac{y ^{2}}{b ^{2}} = 1 subtend an angle of \frac{π}{3} at the centre of the hyperbola. If b^{2} is equal to \frac{l}{m} (1 + n), where l and m are co-prime numbers, then l^{2} + m^{2} + n^{2} is equal to________$ 298. $Let the hyperbola H : \frac{x ^{2}}{a ^{2}} - y^{2} = 1 and the ellipse E : 3 x^{2} + 4 y^{2} = 12 be such that the length of latus rectum of H is equal to the length of latus rectum of E. If e_{H} and e_{E} are the eccentricities of H and E respectively, then the value of 12 (e_{H}^{2} + e_{E}^{2}) is equal to_____.$ 299. $Let H_{1} : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 and H_{2} : - \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1 be two hyperbolas having length of latus rectums 152 and 125 respectively. Let their ecentricities be e_{1} = \frac{5}{2} and e_{2} respectively. If the product of the lengths of their transverse axes is 10010, then 25 e_{2}^{2} is equal to .$ 300. $For some θ \in (0, \frac{π}{2}), let the eccentricity and the length of the latus rectum of the hyperbola x^{2} - y^{2} sec^{2} θ = 8 be e_{1} and l_{1}, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x^{2} sec^{2} θ + y^{2} = 6 be e_{2} and l_{2}, respectively. If e_{1}^{2} = e_{2}^{2} (sec^{2} θ + 1), then (\frac{l _{1} l _{2}}{e _{1} e _{2}}) tan^{2} θ is equal to____.$