Q. The normal to a curve at $P(x,\ y)$ meets the $x$-axis at$G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a
A) Ellipse
B) Parabola
C) Circle
D) Hyperbola
Explanation
(d)
Let the equation of normal is
$Y - y = - \frac{dx}{dy}(X - x)$
It meets the $x$-axis at$G$. Therefore, coordinates of $G$ are $\left( x + y\frac{dy}{dx},\ 0 \right)$
According to given condition,
$x + y\frac{dy}{dx} = 2x\ \ \ \ \ \Rightarrow \ \ y\ dy = x\ dx$
On integrating, we get
$\frac{y^{2}}{2} = \frac{x^{2}}{2} + c\ \ \ \ \Rightarrow \ \ \ \ \ \ \ \ x^{2} - y^{2} = 2c$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.