CircleHardBloom L3
Question
The complete set of real values of a for which at least one tangent to the parabola $y^{2} = 4ax$ becomes normal to the circle $x^{2} + y^{2} - 2ax - 4ay + 3a^{2} = 0$ is:
Options
A.$\lbrack 1,2\rbrack$
B.$\lbrack\sqrt{2},3\rbrack$
C.$R$
D.$R - \{ 0\}$
Solution
[Key Concept: Require at least one tangent condition; reduce to a quadratic in the parameter and bound it.]
Let equation of tangent to $y^{2} = 4ax$ be
$$ty = x + at^{2}$$
If passes through $(a,2a)$
$$\begin{matrix} & \ \Rightarrow \ 2at = a + at^{2} \Rightarrow a(t - 1)^{2} = 0 \\ & \ \Rightarrow \ a \in R - \{ 0\} \end{matrix}$$
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