Question

If the normal at $t_1$ to the hyperbola $xy=c^2$ meets the hyperbola at $t_2$. then find the relation between $t_1$ and $t_2$.

Answer 1

$xy=c^2$ is the equation of hyperbola.

Let $t_1=(x_1,y_1)$ and $t_2=(x_2,y_2)$

slope of normal at $t_1=-\frac{d{x}_t}{d{y}_t}$

$=-\frac{d}{d{y}_1}\left(\frac{c^2}{y_1}\right)$

$=\frac{c^2}{y_1^2}$

Equation of normal $\Rightarrow$

$y-y_1=\frac{c^2}{y_1^2}(x-x_1)$

Not $T_2$ hyperbola satisfies this equation.

$\Rightarrow y_2-y_1=\frac{c^2}{y_1^2}(x_2-x_1)$

Now, Since both are points on hyperbola,

$\Rightarrow x_2=\frac{c^2}{y_2},x_1\frac{c^2}{y_1}$

$\Rightarrow y_2-y_1=\frac{c^2}{y_1^2}.c^2(\frac{1}{y^2}-\frac{1}{y^1})$

$\Rightarrow (y_2-y_2)=\frac{-c^4}{y_1^2}\frac{(y_2-y_1)}{y_2{y}_1}$

$\Rightarrow y_1^3{y}_2=-c^4$

This is the relation between y-coordinates of $t_1$ anf $t_2$.