ParabolaHardBloom L2
Question
Length of the intercept on the normal at the point $P\left( at^{2},2at \right)$ of the parabola $y^{2} = 4ax$ made by the circle described on the focal distance of the point $P$ as diameter is :
Options
A.$a\sqrt{2 + t^{2}}$
B.$\frac{a}{2}\sqrt{1 + t^{2}}$
C.$2a\sqrt{1 + t^{2}}$
D.$a\sqrt{1 + t^{2}}$
Solution
Sol. [Key Concept: The circle on the focal radius $SP$ as diameter passes through the vertex tangent; its intercept along the normal at $P$ comes from the right angle in a semicircle.]
$PRSQ$ is a rectangle
$$\begin{matrix} \Rightarrow \ & PR = SQ = \sqrt{a^{2} + a^{2}t^{2}} \\ & PR = a\sqrt{1 + t^{2}} \end{matrix}$$
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