ParabolaHardBloom L3
Question
$C$ is the centre of the circle with centre $(0,1)$ and radius unity. $P$ is the parabola $y = ax^{2}$. The set of values of ' $a$ ' for which they meet at a point other than the origin, is :
Options
A.$a > 0$
B.$a \in \left( 0,\frac{1}{2} \right)$
C.$\left( \frac{1}{4},\frac{1}{2} \right)$
D.$\left( \frac{1}{2},\infty \right)$
Solution
Sol. [Key Concept: Substitute $y = {ax}^{2}$ into the circle; a second intersection needs the quadratic in $x^{2}$ to have a positive root — this holds for all $a > 0$.]
Put $x^{2} = \frac{y}{a}$
$${\Rightarrow y^{2} + \left( \frac{1}{a} - 2 \right)y = 0 }{\Rightarrow \ y = 0,\ y = 2 - \frac{1}{a} }{2 > 2 - \frac{1}{a} > 0 \Rightarrow a > \frac{1}{2}}$$
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