91Ó°ÊÓ

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