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 abscissa of $P$, then the curve is :

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

Answer 1

Answer: (A), (D)

an ellipse
a hyperbola

Equation of normal $Y-y=-\frac{dy}{dx}(X-x)$
$\Rightarrow G=(x+y\frac{dy}{dx},0)$
According to question
$|x+y\frac{dy}{dx}|=\left|2x\right|\Rightarrow y\frac{dy}{dx}=x$ or $y\frac{dy}{dx}=-3x$
$\Rightarrow ydy=xdx$ or $ydy=-3xdx$
$\Rightarrow \frac{y^2}{2}=\frac{x^2}{2}+c$ or $\frac{y^2}{2}=-\frac{3x^2}{2}+c$
$\Rightarrow x^2-y^2=-2c$ or $3x^2+y^2=2c$