Question

$C$ is the centre of the circle with centre $(0,1)$ and radius unity. $P$ is the parabola $y=a x^{2}$. The set of values of $a$ for which they meet at a point other the the origin is

• A. $a > 0$
• B. $a \in(0,1 / 2)$
• C. $(1 / 4,1 / 2)$
• D. $(1 / 2, \infty)$

Answer 1

Answer: (D)

Putting $x^{2}=y / a$ in the circle $x^{2}+(y-1)^{2}=1$,
we get $\frac{y}{a}+y^{2}-2 y=0$
(Note that for $a<0$ they cannot intersect other than the origin)
Hence we get
$y=0 $ or $ y=2-\frac{1}{a}$
Substituting
$y=2-\frac{1}{a} $ in $ y=a x^{2}$, we get
$a x^{2}=2-\frac{1}{a}$
or $ x^{2}=\frac{2 a-1}{a^{2}} > 0 $
or $ a > \frac{1}{2}$