Dr. SK Goyal Solutions for Chapter: Hyperbola, Exercise 2: INTRODUCTORY EXERCISE 7.2

A common tangent to $9x^2-16y^2=144$ and $x^2+y^2=9$, is

The tangents from $\left(1,2\sqrt{2}\right)$ to the hyperbola $16x^2-25y^2=400$ include between them an angle equal to

The equation of the chord of hyperbola $25x^2-16y^2=400$ whose mid-point is $\left(5, 3\right)$, is

The value of $m$ for which $y=mx+6$ is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$ is

$P$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, N$ is the foot of the perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of the hyperbola, then $OT·ON$ is equal to

A line through the origin meets the circle $x^2+y^2=a^2$ at $P$ and the hyperbola $x^2-y^2=a^2$ at $Q$. Prove that the locus of the point of intersection of tangent at $P$ to the circle with the tangent at $Q$ to the hyperbola is the curve.

Normals are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at the points $P\left(a\sec {\theta }_1,b\tan {\theta }_1\right)$and $Q\left(a\sec {\theta }_2,b\tan {\theta }_2\right)$ meeting the conjugate axis at $G_1$ and $G_2$ respectively. If${\theta }_1+{\theta }_2=\frac{\pi }{2}$, then prove that $C{G}_1.C{G}_2=\frac{a^2{e}^4}{e^2-1}$, where $C$ is the centre of the hyperbola and $e$ is its eccentricity.

Chords of the hyperbola, $x^2-y^2=a^2$ touch the parabola, $y^2=4ax$. Prove that the locus of their middle points is the curve, $y^2(x-a)=x^3$