ParabolaHardBloom L3
Question
If the normal to the parabola $y^{2} = 4ax$ at the point $\left( at^{2},2at \right)$ cuts the parabola again at $\left( aT^{2},2aT \right)$, then complete set of values of $T$ satisfies :
Options
A.$T^{2} \geq 8$
B.$T \in ( - \infty, - 8) \cup (8,\infty)$
C.$- 2 \leq T \leq 2$
D.$T^{2} < 8$
Solution
Sol. [Key Concept: For a normal chord at $t$, the far parameter is $T = - t - \frac{2}{t}$; by AM–GM $|T| \geq \ 2\sqrt{2}$, i.e. $T^{2} \geq \ 8$.]
$$T = - t - \frac{2}{t} $$
$$\begin{matrix} & \ |T| \geq 2\sqrt{2}\ \text{~}\text{(applying A.M.}\text{~} \geq \text{~}\text{G.M.)}\text{~} \\ & T^{2} \geq 8 \end{matrix}$$
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