Question

Find equation of tangent & normal at a point $(p,q)$ on $x^2/a^2-y^2/b^2=1$.

Answer 1

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Slope of tangent

$\frac{2x}{a^2}-\frac{2y}{b^2}\frac{dy}{dx}=0\frac{dy}{dx}=\frac{x}{y}\frac{b^2}{a^2}{\frac{dy}{dx}|}_{(p,q)}=\frac{p{b}^2}{q{a}^2}$

Equation of tangent

$(y-q)=\frac{p{b}^2}{q{a}^2}(x-p)p{b}^{2}x-q{a}^{2}y+q^2{a}^2-p^2{b}^2=0$

Slope of Normal $=\frac{-q{a}^2}{p{b}^2}$

Equation of Normal

$(y-q)=\frac{-q{a}^2}{p{b}^2}(x-p)q{a}^{2}x+p{b}^{2}y+pq(b^2-a^2)=0$