Question

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 abscicca of $P$, then the curve is a:

• A. circle
• B. hyperbola
• C. ellipse
• D. parabola

Answer 1

Answer: (B), (C)

ellipse
hyperbola

Equation of normal at $P(x,y)$ is given by,
$Y-y=-\frac{dx}{dy}(X-x)$
$\Rightarrow G=(x+y\frac{dy}{dx},0)$
Now given condition is,
$|x+y\frac{dy}{dx}|=\left|2x\right|$
$\Rightarrow y\frac{dy}{dx}=x$ or $y\frac{dy}{dx}=-3x$
$\Rightarrow ydy=xdx$ or $ydy=-3xdx$
After integrating,we get
$\frac{y^2}{2}=\frac{x^2}{2}+c$ or $\frac{y^2}{2}=-\frac{3x^2}{2}+c$
$\Rightarrow x^2-y^2=-2c$ or
$3x^2+y^2=2c$
$\Rightarrow x^2-y^2=c_1$ or $3x^2+y^2=c_2$