Practicing Success
The orthogonal trajectories of the family of curves y = cxk are given by |
x2 + cy2 = constant x2 + ky2 = constant kx2 + y2 = constant x2 - ky2 = constant |
x2 + ky2 = constant |
Differentiating the given relation we have, $\frac{dy}{dx}= ckx^{k-1} ⇒ c =\frac{1}{k} x^{1-k }\frac{dy}{dx}$ Putting this value of c 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 $y = -\frac{1}{k} x\frac{dx}{dy}$ ⇒ ky dy + x dx = 0 $⇒ ky^2 + x^2$ = constant. Hence (B) is the correct answer. |