coordinate geometry in polar coordinate Let $G= \{(x, f(x)) \mid x \text{ lies between } 0 \text{ and } 1 \}$
Let $(1,0)$ belong to $G$.
It is given that tangent vector to $G$ at any point is perpendicular to radius vector at that point. 
Is $G$ parabola or ellipse? 

Well, you are given that $(1,f'(x)) \perp (x,f(x))$ and $f(1) = 0$. Thus, $x + f'(x)f(x) = 0$. We can rewrite this as
$$ \frac{d}{dx} f^2(x) = -2x. $$
Integrating, we get $f^2(x) = -x^2 + C$ and by plugging in $f(1) = 0$ we see that $C = 1$. Thus, $f(x) = \pm \sqrt{1 - x^2}$ and thus $G$ is (part of) a circle.
