Question

The orthogonal trajectories for the family of curves $y=\frac{a}{x}$ for $a\ne 0$ are:

• A. Circle
• B. Ellipse
• C. Hyperbola
• D. A pair of straight lines

Answer 1

Answer: (C), (D)

Hyperbola
A pair of straight lines

$y=\frac{a}{x}⟹xy=a$

Differentiating w.r.t. $x$, we get

$x\frac{dy}{dx}+y=0⟹\frac{dy}{dx}=-\frac{y}{x}$

Replacing $\frac{dy}{dx}$ with $-\frac{dx}{dy}$, we get

$-\frac{dx}{dy}=-\frac{y}{x}⟹ydy=xdx$

Integrating on both sides, we get

$\int ydy=\int xdx⟹\frac{y^2}{2}=\frac{x^2}{2}+C_1$

or $y^2-x^2=C$

Hence, for $C\ne 0$, the orthogonal trajectories are rectangular hyperbola and for $C=0$, the orthogonal trajectories are the lines $x=\pm y$