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 $\frac{y}{a}+y^2-2y=0$
(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 $x^2=\frac{2a-1}{a^2}>0$
or $a>\frac{1}{2}$