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

Answer 1

Answer: (C)

Slope of tangent $=\frac{dy}{dx}$
$\therefore$ Slope of normal $=-\frac{dx}{dy}$
Thus, the equation of normal is
$Y-y=-\frac{dx}{dy}(X-x)$
This meets x-axis (y = 0), where
$-y=-\frac{dx}{dy}(X-x)$ or $X=x+y\frac{dy}{dx}$
$\therefore G$ is $(x+y\frac{dy}{dx},0)$
$\therefore OG=2x$
$\therefore x+y\frac{dy}{dx}=2x$
or $y\frac{dy}{dx}=x$ or $ydy=xdx$
Integrating, we get $\frac{y^2}{2}=\frac{x^2}{2}+\frac{C}{2}$
or $y^2-x^2=c$, which is a hyperbola