# Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on.

$H$: $xy=c^2$ is a hyperbola

(i) Show that $H$ can be represented by the parametric equations $x=ct$ , $y= \frac{c}{t}$.

If we take $y= \frac{c}{t}$ and rearrange it to $t= \frac{c}{y}$ and subbing this into $x=ct$

$$x = c(\frac{c}{y})$$

$$\therefore xy = c^2$$

(ii) Find the gradient of the normal to $H$ at the point $T$ with the coordinates $(ct, \frac{c}{t})$

As $xy = c^2$

$$\Leftrightarrow y = c^2 x^{-1}$$

$$\Leftrightarrow y' = -\frac{c^2}{x^2}$$

$$\Leftrightarrow y'_{x=ct} = \frac{-1}{t^2}$$

Hence gradient of the normal is $m = t^2$

The normal to H at the point $T$ meets $H$ again at $P(cp , \frac{c}{p})$

(iiI) By finding the gradient of the line TP and comparing it with the gradient from (ii), find a relationship between $t$ and $p$

So repeating steps of (ii) again

$xy = c^2$

$$\Leftrightarrow y = c^2 x^{-1}$$

$$\Leftrightarrow y' = -\frac{c^2}{x^2}$$

$$\Leftrightarrow y'_{x=cp} = \frac{-1}{p^2}$$