Question

Find whether the curves $x^2+4y^2=8$ and $x^2-2y^2=4$ are orthogonal or not.

Answer 1

$f\left(x\right)=x^2+4y^2=8$, $x^2-2y^2=4$

$f^{\prime }\left(x\right)=2x+8y\frac{dy}{dx}=0$, $2x-4y\frac{dy}{dx}=0$

$\frac{dy}{dx}=\frac{-2x}{8y}$

$M_1=\frac{-1}{4}\left(\frac{x}{y}\right)$

$M_2=\frac{dy}{dx}=\frac{x}{2y}$

If $M-1M_2=-1$ (orthogonal)

$\therefore$ They are not orthogonal.