Embibe Experts Solutions for Chapter: Hyperbola, Exercise 1: Exercise 1

If $\theta$ is the angle between the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with eccentricity $e,$ then $\sec \frac{\theta }{2}$ can be-

If $m_1$ and $m_2$ are the slopes of the tangents to the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ which passes through the point $\left(6,2\right),$ then

The foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. If the eccentricity of the hyperbola is $2$, then the equation of the tangent to this hyperbola passing through the point $\left(4,6\right)$ is

If the tangent and the normal to the hyperbola $x^2-y^2=4$ at a point cut off intercepts $a_1$ and $a_2$, respectively on the $X$-axis, and $b_1$ and $b_2$, respectively on the $Y$-axis, then the value of $a_1{a}_2+b_1{b}_2$ is

Let $P\left(a\mathrm{sec\theta }, b\mathrm{tan\theta }\right)$ and $Q\left(a\sec \varphi , b\tan \varphi \right)$ where $\theta +\varphi =\frac{\mathrm{\pi }}{2},$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If $\left(h, k\right)$ is the point of intersection of the normals at $P$ and $Q,$ then $k$ is equal to-

If chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax.$ Then, the locus of their middle point is

The locus of the foot of the perpendicular from the centre fo the hyperbola $xy=c^2$ on a variable tangent is:

If a real circle will pass through the points of intersection of hyperbola $x^2-y^2=a^2$ and parabola $y=x^2$, then-