Hyperbola PYQs - Last 5 Years

The normal to the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1\) at the point \(\left( {8,3\sqrt 3 } \right)\) on it passes through the point :

Let the tangent drawn to the parabola \(y^{2}=24 x\) at the point \((\alpha, \beta)\) is perpendicular to the line \(2 x+2 y=5\). Then the normal to the hyperbola \(\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1\) at the point $$(\alpha...

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 :

Let the eccentricity of the hyperbola \({{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1\) be \({5 \over 4}\). If the equation of the normal at the point \(\left( {{8 \over {\sqrt {5} }},{{12} \over {5}}} \right)\) on the hyperbola is ...

Let the equation of two diameters of a circle \(x^{2}+y^{2}-2 x+2 f y+1=0\) be \(2 p x-y=1\) and \(2 x+p y=4 p\). Then the slope m \(\in\) \((0, \infty)\) of the tangent to the hyperbola \(3 x^{2}-y^{2}=3\) passing through the centre of t...

Let the foci of the ellipse \(\frac{x^{2}}{16}+\frac{y^{2}}{7}=1\) and the hyperbola \(\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}\) coincide. Then the length of the latus rectum of the hyperbola is :

Let the hyperbola \(H:{{{x^2}} \over {{a^2}}} - {y^2} = 1\) 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 eccentriciti...