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

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

Answer 1

The correct option is D hyperbola

Equation of normal to the given curve is $Y-y=-\frac{dx}{dy}(X-x)$
Solving with x-axis, i.e. Y = 0
$-y=-\frac{dx}{dy}(X-x)X-x=\frac{dy}{dx}yX=x+\frac{dy}{dx}y\therefore G=[x+\frac{dy}{dx}y,0]$
Given $OG=2x\therefore x+\frac{dy}{dx}y=2x\Rightarrow y.dy=xdx\Rightarrow \frac{y^2}{2}+c=\frac{x^2}{2}\therefore x^2-y^2=2c$ which is hyperbola.