Show that the curves $4x=y^2$ and $4xy=k$ cut at right angle if $k^2=512$
Since the curves cut at right angle $\implies$ their tangents are perpendicular to each other
Consider $4x=y^2$
Differentiating w.r.t x gives
$4= 2y \dfrac{dy}{dx}$
$\implies$ $\dfrac{dy}{dx}=\dfrac{2}{y}$
Consider $4xy=k$
Differentiating w.r.t x gives
$4\left(x\dfrac{dy}{dx}+y\right)=0$
$\implies$ $\dfrac{dy}{dx}=-\dfrac{y}{x}$
Since the tangents are perpendicular to each other
$\implies$ $\dfrac{2}{y}\times \left(\dfrac{-y}{x}\right)=-1$ $\implies$ $x=2$
Substituting the value of x in the equation of the curve $4xy=k$ gives
$8y=k$ $\implies$ $y = \dfrac{k}{8}$
Substituting the values of x and y in the equation of the curve $4x=y^2$ gives
$4 \times 2 = \left(\dfrac{k}{8}\right)^2$
$\implies$ $k^2=8\times 64 = 512$
$\therefore$ $\;$ The two curves cut at right angle if $k^2 = 512$