Question

The differential equation of family of parabola with foci at the origin and axis along the x axes :

• A. y$\left( \frac{dy}{dx} \right)^{2}$ + 2x$\frac{dy}{dx}$ – y = 0
• B. x$\left( \frac{dy}{dx} \right)^{2}$ + 2y$\frac{dy}{dx}$ – y = 0
• C. y$\left( \frac{dy}{dx} \right)^{2}$+ 2x $\frac{dy}{dx}$ + y = 0
• D. None of these

Answer 1

Answer: (A)

y$\left( \frac{dy}{dx} \right)^{2}$ + 2x$\frac{dy}{dx}$ – y = 0

Answer 2

Parabola

distance form focus = distance from directrix

x² + y² = (2a + x)²

y² = 4a(a + x) . .........(i)

2y$\frac{dy}{dx}$ = 4a(0 + 1)

a = $\frac{dy}{dx}$

using (i) y² = 2y $\left( \frac{y}{2}\frac{dy}{dx} + x \right)$

y$\left( \frac{dy}{dx} \right)^{2}$ + 2x$\frac{dy}{dx}$– y = 0