Question

Find the curve for which the length of normal is equal to the radius vector –

• A. Circle
• B. Hyperbola
• C. Ellipse
• D. Circle or equilateral hyperbola

Answer 1

Answer: (D)

Circle or equilateral hyperbola

Answer 2

Here radius vector = OP

and length of normal = PN

according to question PN = OP

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

Ž y²$\left( 1 + \left( \frac{dy}{dx} \right)^{2} \right)$ = x² + y² Ž y² $\left( \frac { d y } { d x } \right) ^ { 2 }$ = x²

or ± y $\frac{dy}{dx}$ = x or x dx ± y dy = 0

or 2x dx ± 2y dy = 0

integrating, we get x² ± y² = c²

This equation represents a circle or equilateral hyperbola as + ve or – ve sign be taken