Question

The orthogonal trajectories of the family of curves $xy=k^2$ are:

• A. rectangular hyperbola
• B. $y^2=4a(x+a)$
• C. $x^2-y^2=a^2$
• D. system of circles with centre on x-axis

Answer 1

Answer: (A), (C)

rectangular hyperbola
$x^2-y^2=a^2$

$xy=k^2\Rightarrow y+x\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{y}{x}$
Replacing $\frac{dy}{dx}\to -\frac{dx}{dy}$
$\therefore -\frac{dx}{dy}=-\frac{y}{x}\Rightarrow xdx=ydy\Rightarrow x^2=y^2+c\Rightarrow x^2-y^2=c$