Chapter 11

If the tangent at a point on the ellipse \(\frac{x^{2}}{27}+\frac{y^{2}}{3}=1\) meets the coordinate axes at A and B, and \(O\) is the origin, then the minimum area (in sq. units) of the triangle \(\mathrm{OAB}\) is : (a) \(3 \sqrt{3}\) (b) \(\frac{9}{2}\) (c) 9 (d) \(\frac{9}{\sqrt{3}}\)

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{5}=1\), is: (a) \(\frac{27}{2}\) (b) 27 (c) \(\frac{27}{4}\) (d) 18

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is: [Online April 11, 2015] (a) \(\frac{2 \sqrt{2}-1}{2}\) (b) \(\sqrt{2}-1\) (c) \(\frac{1}{2}\) (d) \(\frac{\sqrt{2}-1}{2}\)

The locus of the foot of perpendicular drawn from the centre of theedlipse \(x^{2}+3 y^{2}=6\) on any tangent to it is (a) \(\left(x^{2}+y^{2}\right)^{2}=6 x^{2}+2 y^{2}\) (b) \(\left(x^{2}+y^{2}\right)^{2}=6 x^{2}-2 y^{2}\) (c) \(\left(x^{2}-y^{2}\right)^{2}=6 x^{2}+2 y^{2}\) (d) \(\left(x^{2}-y^{2}\right)^{2}=6 x^{2}-2 y^{2}\)

A stair-case of length \(l\) rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio \(1: 2\). If the stair-case begins to slide on the floor, then the locus of P is: (a) an ellipse of eccentricity \(\frac{1}{2}\) (b) an ellipse of eccentricity \(\frac{\sqrt{3}}{2}\) (c) a circle of radius \(\frac{1}{2}\) (d) a circle of radius \(\frac{\sqrt{3}}{2} l\)

If \(\mathrm{OB}\) is the semi-minor axis of an ellipse, \(\mathrm{F}_{1}\) and \(\mathrm{F}_{2}\) are its foci and the angle between \(\mathrm{F}_{1} \mathrm{~B}\) and \(\mathrm{F}_{2} \mathrm{~B}\) is a right angle, then the square of the eccentricity of the ellipse is: (a) \(\frac{1}{2}\) (b) \(\frac{1}{\sqrt{2}}\) (c) \(\frac{1}{2 \sqrt{2}}\) (d) \(\frac{1}{4}\)

The equation of the circle passing through the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\), and having centre at \((0,3)\) is \([\mathbf{2 0 1 3}]\). (a) \(x^{2}+y^{2}-6 y-7=0\) (b) \(x^{2}+y^{2}-6 y+7=0\) (c) \(x^{2}+y^{2}-6 y-5=0\) (d) \(x^{2}+y^{2}-6 y+5=0\)

A point on the ellipse, \(4 x^{2}+9 y^{2}=36\), where the normal is parallel to the line, \(4 x-2 y-5=0\), is : (a) \(\left(\frac{9}{5}, \frac{8}{5}\right)\) (b) \(\left(\frac{8}{5},-\frac{9}{5}\right)\) (c) \(\left(-\frac{9}{5}, \frac{8}{5}\right)\) (d) \(\left(\frac{8}{5}, \frac{9}{5}\right)\)

Let the equations of two ellipses be \(E_{1}: \frac{x^{2}}{3}+\frac{y^{2}}{2}=1\) and \(E_{2}: \frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1\), If the product of their eccentricities is \(\frac{1}{2}\), then the length of theminor axis of ellipse \(E_{2}\) is : (a) 8 (b) 9 (c) 4 (d) 2

Equation of the line passing through the points of intersection of the parabola \(x^{2}=8 y\) and the ellipse \(\frac{x^{2}}{3}+y^{2}=1\) is: \(\quad\) [Online April 9, 2013] (a) \(y-3=0\) (b) \(y+3=0\) (c) \(3 y+1=0\) (d) \(3 y-1=0\)

If \(P_{1}\) and \(P_{2}\) are two points on the ellipse \(\frac{x^{2}}{4}+y^{2}=1\) at which the tangents are parallel to the chord joining the points \((0,1)\) and \((2,0)\), then the distance between \(P_{1}\) and \(P_{2}\) is [Online May 12, 2012] (a) \(2 \sqrt{2}\) (b) \(\sqrt{5}\) (c) \(2 \sqrt{3}\) (d) \(\sqrt{10}\)

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point \((-3,1)\) and has eccentricity \(\sqrt{\frac{2}{5}}\) is [2011] (a) \(5 x^{2}+3 y^{2}-48=0\) (b) \(3 x^{2}+5 y^{2}-15=0\) (c) \(5 x^{2}+3 y^{2}-32=0\) (d) \(3 x^{2}+5 y^{2}-32=0\)

The ellipse \(x^{2}+4 y^{2}=4\) is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point \((4,0)\). Then the equation of the ellipse is : (a) \(x^{2}+12 y^{2}=16\) (b) \(4 x^{2}+48 y^{2}=48\) (c) \(4 x^{2}+64 y^{2}=48\) (d) \(x^{2}+16 y^{2}=16\)

A focus of an ellipse is at the origin. The directrix is the line \(x=4\) and the eccentricity is \(\frac{1}{2}\). Then the length of the semi-major axis is (a) \(\frac{8}{3}\) (b) \(\frac{2}{3}\) (c) \(\frac{4}{3}\) (d) \(\frac{5}{3}\)

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is [2006] (a) \(\frac{3}{5}\) (b) \(\frac{1}{2}\) (c) \(\frac{4}{5}\) (d) \(\frac{1}{\sqrt{5}}\)

An ellipse has \(O B\) as semi minor axis, \(F\) and \(F^{\prime}\) its focii and the angle \(F B F^{\prime}\) is a right angle. Then the eccentricity of the ellipse is (a) \(\frac{1}{\sqrt{2}}\) (b) \(\frac{1}{2}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{\sqrt{3}}\)

The eccentricity of an ellipse, with its centre at the origin, is \(\frac{1}{2}\). If one of the directrices is \(x=4\), then the equation of the ellipse is: [2004] (a) \(4 x^{2}+3 y^{2}=1\) (b) \(3 x^{2}+4 y^{2}=12\) (c) \(4 x^{2}+3 y^{2}=12\) (d) \(3 x^{2}+4 y^{2}=1\)

If the line \(y=m x+c\) is a common tangent to the hyperbola \(\frac{x^{2}}{100}-\frac{y^{2}}{64}=1\) and the circle \(x^{2}+y^{2}=36\), then which one of the following is true? \(\quad\) [Sep. 05, 2020 (II)] (a) \(c^{2}=369\) (b) \(5 m=4\) (c) \(4 c^{2}=369\) (d) \(8 m+5=0\)

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 \(\left(a^{2}, e^{2}\right)\) is equal to: (a) \(\left(\frac{9}{2}, 3\right)\) (b) \(\left(\frac{3}{2}, 2\right)\) (c) \(\left(\frac{9}{2}, 2\right)\) (d) \((9,3)\)

Let \(e_{1}\) and \(e_{2}\) be the eccentricities of the ellipse, \(\frac{x^{2}}{25}+\frac{y^{2}}{b^{2}}=1(b<5)\) and the hyperbola, \(\frac{x^{2}}{16}-\frac{y^{2}}{b^{2}}=1\) respectively satisfying \(e_{1} \mathrm{e}_{2}=1\). If \(\alpha\) and \(\beta\) are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair \((\alpha, \beta)\) is equal to : (a) \((8,12)\) (b) \(\left(\frac{20}{3}, 12\right)\) (c) \(\left(\frac{24}{5}, 10\right)\) (d) \((8,10)\)

A line parallel to the straight line \(2 x-y=0\) is tangent to the hyperbola \(\frac{x^{2}}{4}-\frac{y^{2}}{2}=1\) at the point \(\left(x_{1}, y_{1}\right)\). Then \(x_{1}^{2}+5 y_{1}^{2}\) is equal to : (a) 6 (b) 8 (c) 10 (d) 5

For some \(\theta \in\left(0, \frac{\pi}{2}\right)\), if the eccentricity of the hyperbola, \(x^{2}-y^{2} \sec ^{2} \theta=10\) is \(\sqrt{5}\) times the eccentricity of the ellipse, \(x^{2} \sec ^{2} \theta+y^{2}=5\), then the length of the latus rectum of the ellipse, is: (a) \(2 \sqrt{6}\) (b) \(\sqrt{30}\) (c) \(\frac{2 \sqrt{5}}{3}\) (d) \(\frac{4 \sqrt{5}}{3}\)

Let \(\mathrm{P}\) 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 \(S^{\prime}\) denote the foci of the hyperbola where \(S\) lies on the positive \(x\)-axis then P divides \(\mathrm{SS}^{\prime}\) in a ratio: [April 12, 2019 (I)] (a) \(13: 11\) (b) \(14: 13\) (c) \(5: 4\) (d) \(2: 1\)

The equation of a common tangent to the curves, \(y^{2}=16 x\) and \(x y=-4\), is : [April 12, 2019 (II)] (a) \(x-y+4=0\) (b) \(x+y+4=0\) (c) \(x-2 y+16=0\) (d) \(2 x-y+2=0\)

If \(5 x+9=0\) is the directrix of the hyperbola \(16 x^{2}-9 y^{2}=144\), then its corresponding focus is : [April 10, 2019 (II)] (a) \((5,0)\) (b) \(\left(-\frac{5}{3}, 0\right)\) (c) \(\left(\frac{5}{3}, 0\right)\) (d) \((-5,0)\)

If the line \(y=m x+7 \sqrt{3}\) is normal to the hyperbola \(\frac{x^{2}}{24}-\frac{y^{2}}{18}=1\), then a value of \(m\) is: \(\quad\) [April 09, 2019 (I)] (a) \(\frac{\sqrt{5}}{2}\) (b) \(\frac{\sqrt{15}}{2}\) (c) \(\frac{2}{\sqrt{5}}\) (d) \(\frac{3}{\sqrt{5}}\)

If the vertices of a hyperbola be at \((-2,0)\) and \((2,0)\) and one of its foci be at \((-3,0)\), then which one of the following points does not lie on this hyperbola? (a) \((-6,2 \sqrt{10})\) (b) \((2 \sqrt{6}, 5)\) (c) \((4, \sqrt{15})\) (d) \((6,5 \sqrt{2})\)

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 , then the eccentricity of the hyperbola is: (a) \(\frac{13}{12}\) (b) 2 (c) \(\frac{13}{6}\) (d) \(\frac{13}{8}\)

Let the length of the latus rectum of an ellipse with its major axis along \(x\)-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it? [Jan. 11, 2019 (II)] (a) \((4 \sqrt{2}, 2 \sqrt{2})\) (b) \((4 \sqrt{3}, 2 \sqrt{2})\) (c) \((4 \sqrt{3}, 2 \sqrt{3})\) (d) \((4 \sqrt{2}, 2 \sqrt{3})\)

The equation of a tangent to the hyperbola \(4 x^{2}-5 y^{2}=20\) parallel to the line \(x-y=2\) is: \(\mid\) Jan 10,2019 (I)] (a) \(x-y+1=0\) (b) \(x-y+7=0\) (c) \(x-y+9=0\) (d) \(x-y-3=0\)

Let \(\mathrm{S}=\left\\{(x, y) \in \mathbf{R}^{2}: \frac{y^{2}}{1+\mathrm{r}}-\frac{x^{2}}{1-\mathrm{r}}=1\right\\}\) where \(\mathrm{r} \neq \pm 1\) Then S represents: [Jan. 10, 2019 (II)] (a) a hyperbola whose eccentricity is \(\frac{2}{\sqrt{1-r}}\), when \(0<\mathrm{r}<1\) (b) an ellipse whose eccentricity is \(\sqrt{\frac{2}{r+1}}\), when \(r>1\) (c) a hyperbola whose eccentricity is \(\frac{2}{\sqrt{r+1}}\), when \(01\)

Let \(0<\theta<\frac{\pi}{2}\). If the eccentricity of the hyperbola \(\frac{x^{2}}{\cos ^{2} \theta}-\frac{y^{2}}{\sin ^{2} \theta}=1\) is greater than 2 , then the length of its latus rectum lies in the interval: (a) \((3, \infty)\) (b) \((3 / 2,2]\) (c) \((2,3]\) (d) \((1,3 / 2]\)

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 : (a) \(\frac{3}{2}\) (b) \(\sqrt{3}\) (c) 2 (d) \(\frac{2}{\sqrt{3}}\)

The locus of the point of intersection of the lines, \(\sqrt{2} x-y+4 \sqrt{2} k=0\) and \(\sqrt{2} k x+k y-4 \sqrt{2}=0(k\) is any non-zero real parameter) is. \(\quad\) Online April 16, 2018] (a) A hyperbola with length of its transverse axis \(8 \sqrt{2}\) (b) An ellipse with length of its major axis \(8 \sqrt{2}\) (c) An ellipse whose eccentricity is \(\frac{1}{\sqrt{3}}\) (d) A hyperbola whose eccentricity is \(\sqrt{3}\)

The locus of the point of intersection of the straight lines, \(\mathrm{tx}-2 \mathrm{y}-3 \mathrm{t}=0\) \(\mathrm{x}-2 \mathrm{ty}+3=0(\mathrm{t} \in \mathrm{R})\), is: (a) an ellipse with eccentricity \(\frac{2}{\sqrt{5}}\) (b) an ellipse with the length of major axis 6 (c) a hyperbola with eccentricity \(\sqrt{5}\) (d) a hyperbola with the length of conjugate axis 3

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : (a) \(\frac{2}{\sqrt{3}}\) (b) \(\sqrt{3}\) (c) \(\frac{4}{3}\) (d) \(\frac{4}{\sqrt{3}}\)

Let a aand \(b\) respectively be the semitransverse and semiconjugate axes of a hyperbola whose eccentricity satisfies the equation \(9 \mathrm{e}^{2}-18 \mathrm{e}+5=0 .\) If \(\mathrm{S}(5,0)\) is a focus and \(5 \mathrm{x}=9\) is the corresponding directrix of this hyperbola, then \(a^{2}-b^{2}\) is equal to: \(\quad\) Online April 9, 2016] (a) \(-7\) (b) \(-5\) (c) \(\underline{5}\) (d) 7

An ellipse passes through the foci of the hyperbola, \(9 x^{2}-4 y^{2}=36\) and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is \(\frac{1}{2}\), then which of the following points does not lie on the ellipse? |Online April \(\mathbf{1 0}\), 2015| (a) \(\left(\sqrt{\frac{13}{2}}, \sqrt{6}\right)\) (b) \(\left(\frac{\sqrt{39}}{2}, \sqrt{3}\right)\) (c) \(\left(\frac{1}{2} \sqrt{13}, \frac{\sqrt{3}}{2}\right)\) (d) \((\sqrt{13}, 0)\)

The tangent at an extremity (in the first quadrant) of latus rectum of the hyperbola \(\frac{x^{2}}{4}-\frac{y^{2}}{5}=1\), meet \(x\)-axis and \(y\)-axis at \(\mathrm{A}\) and \(\mathrm{B}\) respectively. Then \((\mathrm{OA})^{2}-(\mathrm{OB})^{2}\), where \(\mathrm{O}\) is the origin, equals: (a) \(-\frac{20}{9}\) (b) \(\frac{16}{9}\) (c) 4 (d) \(-\frac{4}{3}\)

Let \(\mathrm{P}(3 \sec \theta, 2 \tan \theta)\) and \(\mathrm{Q}(3 \sec \phi, 2 \tan \phi)\) where \(\theta+\phi=\frac{\pi}{2}\), be two distinct points on the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\). Then the ordinate of the point of intersection of the normals at \(\mathrm{P}\) and \(\mathrm{Q}\) is: (a) \(\frac{11}{3}\) (b) \(-\frac{11}{3}\) (c) \(\frac{13}{2}\) (d) \(-\frac{13}{2}\)

A common tangent to the conics \(x^{2}=6 y\) and \(2 x^{2}-4 y^{2}=9\) is: \(\quad\) [Online April 25, 2013] (a) \(x-y=\frac{3}{2}\) (b) \(x+y=1\) (c) \(x+y=\frac{9}{2}\) (d) \(x-y=1\)

A tangent to the hyperbola \(\frac{x^{2}}{4}-\frac{y^{2}}{2}=1\) meets \(x\)-axis at \(\mathrm{P}\) and \(y\)-axis at \(\mathrm{Q}\). Lines \(\mathrm{PR}\) and \(\mathrm{QR}\) are drawn such that OPRQ is a rectangle (where \(\mathrm{O}\) is the origin). Then \(\mathrm{R}\) lies on: [Online April 23, 2013] (a) \(\frac{4}{x^{2}}+\frac{2}{y^{2}}=1\) (b) \(\frac{2}{x^{2}}-\frac{4}{y^{2}}=1\) (c) \(\frac{2}{x^{2}}+\frac{4}{y^{2}}=1\) (d) \(\frac{4}{x^{2}}-\frac{2}{y^{2}}=1\)

If the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1\) coincide with the foci of the hyperbola \(\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25}\), then \(b^{2}\) is equal to (a) 8 (b) 10 (c) 7 (d) 9

If the eccentricity of a hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}=1\), which passes through \((k, 2)\), is \(\frac{\sqrt{13}}{3}\), then the value of \(k^{2}\) is (a) 18 (b) 8 (c) 1 (d) 2

The equation of the hyperbola whose foci are \((-2,0)\) and \((2,0)\) and eccentricity is 2 is given by : (a) \(x^{2}-3 y^{2}=3\) (b) \(3 x^{2}-y^{2}=3\) (c) \(-x^{2}+3 y^{2}=3\) (d) \(-3 x^{2}+y^{2}=3\)

For the Hyperbola \(\frac{x^{2}}{\cos ^{2} \alpha}-\frac{y^{2}}{\sin ^{2} \alpha}=1\), which of the following remains constant when \(\alpha\) varies \(=\) ? (a) abscissae of vertices (b) abscissae of foci (c) eccentricity (d) directrix.

The locus of a point \(P(\alpha, \beta)\) moving under the condition that the line \(y=\alpha x+\beta\) is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) is (a) an ellipse (b) a circle (c) a parabola (d) a hyperbola

The foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1\) and the hyperbola \(\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25}\) coincide. Then the value of \(b^{2}\) is (a) 9 (b) 1 (c) 5 (d) 7