Question:
Find the equation of the tangent and the normal to the following curves at the indicated points:
$x y=c^{2}$ at $(c t, c / t)$
Solution:
finding slope of the tangent by differentiating the curve
$\frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\mathrm{y}}{\mathrm{x}}$
$m($ tangent $)$ at $\left(c t, \frac{c}{t}\right)=-\frac{1}{t^{2}}$
normal is perpendicular to tangent so, $m_{1} m_{2}=-1$
$\mathrm{m}$ (normal) at $\left(\mathrm{ct}, \frac{\mathrm{c}}{\mathrm{t}}\right)=\mathrm{t}^{2}$
equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$
$y-\frac{c}{t}=-\frac{1}{t^{2}}(x-c t)$
equation of normal is given by $y-y_{1}=m($ normal $)\left(x-x_{1}\right)$
$y-\frac{c}{t}=t^{2}(x-c t)$