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

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

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

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

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None of these

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Solution

The correct option is B $b^{2} c = 4 a^{3}$
Let the points of the focal chord be $(a {t_{1}}^{2}, 2 a t_{1})$ and $({a t}_{2}^{2}, 2 a t_{2})$ .

Then $t_{1} t_{2} = - 1$

Equation of the focal chord is $(t_{1} + t_{2}) y = 2 x + 2 a t_{1} t_{2}$

Given $b = \frac{2 a t_{1} t_{2}}{\sqrt{{(t_{1} + t_{2})}^{2} + 4}} = \frac{- 2 a}{\sqrt{2 + {t_{1}}^{2} + {t_{2}}^{2}}}$

Also, $c^{2} = a^{2} {({t_{1}}^{2} - {t_{2}}^{2})}^{2} + 4 a^{2} {(t_{1} - t_{2})}^{2} = a^{2} {(t_{1} - t_{2})}^{2} [{(t_{1} + t_{2})}^{2} + 4]$

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

$\Rightarrow c = a ({t_{1}}^{2} + {t_{2}}^{2} + 2)$

Now, $b^{2} = \frac{4 a^{2}}{({t_{1}}^{2} + {t_{2}}^{2} + 2)} = \frac{4 a^{2}}{c / a} = \frac{4 a^{3}}{c} \Rightarrow b^{2} c = 4 a^{3}$

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