Question

$P$ is a point on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, $N$ is the foot of 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 hyperbola, then the value of $OT\times ON$is

Answer 1

Let $P(x_1,y_1)$
$\therefore N(x_1,0)\Rightarrow ON=x_1\text{units}$
Equation of tangent at $P$ is
$T=0\Rightarrow \frac{x{x}_1}{4}-\frac{y{y}_1}{9}=1$
The point of intersection of this tangent on the transverse axis $(y=0)$ is
$T=(\frac{4}{x_1},0)\Rightarrow OT=\frac{4}{x_1}\text{units}\therefore OT\times ON=\frac{4}{x_1}\times x_1=4$