If the length of a focal chord of the parabola y2=4ax at a distance b from the vertex is c, then

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If the length of a focal chord of the parabola $y^{2} = 4 a x$ at $a$ distance $b$ from the vertex is $c,$ then

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$2 a^{2} = b c$

$a c = b^{2}$

$a^{3} = b^{2} c$

$b^{2} c = 4 a^{3}$

answer is D.

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Detailed Solution

Let $P (a {t_{1}}^{2}, 2 a t_{1})$ and $Q (a {t_{2}}^{2}, 2 a t_{2})$ be the and point of a focal chord of the parabola

$y^{2} = 4 a x$ . Then,

$P Q = a {(t_{2} - t_{1})}^{2}$

The equation of $P Q$ is

$(t_{1} + t_{2}) y = 2 x - 2 a [∵ t_{1} t_{2} = - 1]$

lt is given that $P Q = c,$ and it is at $a$ distance $b$ from the vertex.

$\begin{array}{l} ∴ a {(t_{2} - t_{1})}^{2} = c and, |\frac{- 2 a}{\sqrt{{(t_{1} + t_{2})}^{2} + 4}}| = b \\ \Rightarrow {(t_{2} - t_{1})}^{2} = \frac{c}{a} and, \frac{2 a}{(t_{2} - t_{1})} = b \\ \Rightarrow {(\frac{2 a}{b})}^{2} = \frac{c}{a} \Rightarrow 4 a^{3} = b^{2} c \end{array}$