Applications of Derivatives Question - None

Question

The equation of family of curves for which the length of normal at any point P is equal to the distance of P from origin, is

Accepted Answer

Solution:

Let $P(x, y)$ be the point on the curve.

$$OP = \sqrt{x^2 + y^2}$$

where $OP$ is the radius vector (distance from origin).

$PN$ = Length of normal at point $P$

The length of normal is given by: $$PN = \frac{y}{\left|\frac{dy}{dx}\right|} \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$$

Given that $OP = PN$:

$$\sqrt{x^2 + y^2} = \frac{y}{\left|\frac{dy}{dx}\right|} \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$$

Squaring both sides:

$$x^2 + y^2 = y^2\left[1 + \left(\frac{dy}{dx}\right)^2\right]$$

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

$$\frac{dy}{dx} = \pm\frac{x}{y}$$

Separating variables:

$$y\,dy = \pm x\,dx$$

Integrating:

$$\frac{y^2}{2} = \pm\frac{x^2}{2} + C'$$

$$y^2 = \pm x^2 + C$$

∴ Answer is (b)

Question: