Question

The curve for which the normal at any point $P(x,y)$ and the line joining origin to the point $P$ form an isosceles triangle with $x-$axis as base is

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

Answer 1

Answer: (C), (D)

A circle

It is given that the triangle $OPG$ is an isosceles triangle.
Therefore, $OM=MG=$ sub-normal
$\Rightarrow x=\left|y\frac{dy}{dx}\right|\Rightarrow xdx=\pm ydy$
On intergating both sides, we get $x^2-y^2=C,$ or $x^2+y^2=c$ which is a rectangular hyperbola or a circle.