Question

If the product of the slope of tangent to curve at $(x,y)$ and its y-co-ordinate is equal to the x-co-ordinate of the point, then it represent.

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

Answer 1

The correct option is D rectangular hyperbola

Let slope of a tangent at point $(x,y)$ to the curve is $=\frac{dy}{dx}$
$\therefore y\frac{dy}{dx}=x$ (given)
$\therefore ydy=xdx$
Integrating both side $\int ydy=\int xdx$
$\therefore \frac{y^2}{2}=\frac{x^2}{2}+c^{\prime }$
$\therefore y^2-x^2+2c^{\prime }$
$\therefore y^2-x^2=c$
where $2c^{\prime }=c$
Which is a rectangular hyperbola