Question

The orthogonal trajectories of the family of curves $y=c{x}^k$ are given by:

• A. $x^2+c{y}^2=$ constant
• B. $x^2+k{y}^2=$ constant
• C. $k{x}^2+y^2=$ constant
• D. $x^2-k{y}^2=$ constant

Answer 1

The correct option is B $x^2+k{y}^2=$ constant

$y=c{x}^k$

Differentiating, we have

$\frac{dy}{dx}=ck{x}^{k-1}\Rightarrow c=\frac{1}{k}{x}^{1-k}\frac{dy}{dx}$

Substituting this value in the given equation, we have

$y=\frac{1}{k}{x}^{1-k}\frac{dy}{dx}{x}^k=\frac{1}{k}x\frac{dy}{dx}$

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

$y=-\frac{1}{k}x\frac{dx}{dy}\Rightarrow kydy+xdx=0$

$\Rightarrow k{y}^2+x^2=c$