Question

If the two curves $y^2=4ax$ and $xy=c^2$ cut orthogonally such that $c^4=\lambda a^4,$ then the value of $\lambda$ is

Answer 1

Let curves intersect at $P(x_1,y_1)$ orthogonally.
$y^2=4ax\cdots \left(1\right)$
Differentiating w.r.t. $x,$ we get
$\frac{dy}{dx}=\frac{2a}{y}=m_1$ (say)
$m_1$ at point $P$ is $\frac{2a}{y_1}$

Also, $xy=c^2\cdots \left(2\right)$
Differentiating w.r.t. $x,$ we get
$x\frac{dy}{dx}+y=0$
$\Rightarrow \frac{dy}{dx}=-\frac{y}{x}=m_2$ (say)
$m_2$ at point $P$ is $-\frac{y_1}{x_1}$

According to orthogonal condition at $P,$
$m_1\times m_2=-1$
$\Rightarrow x_1=2a$
From $\left(2\right),$ we have
$x_1^2{y}_1^2=c^4$
$\Rightarrow 4a^2\cdot 4a{x}_1=c^4$
$\Rightarrow 4a^2\cdot 4a\cdot 2a=c^4$
$\Rightarrow c^4=32a^4$