If $b, k$ are intercepts of a focal chord of the parabola $y^{2} = 4 a x$ then $k$ is equal to

\frac{a b}{b - a}

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\frac{b}{b - a}

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\frac{a}{b - a}

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\frac{a b}{a - b}

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Solution

The correct option is A $\frac{a b}{b - a}$
If coordinates of one end focal chord are $P (a t^{2}, 2 a t)$ then the coordinate of other end ill be $Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$ .
$∴ S P = \sqrt{{(a t^{2} - a)}^{2} + {(2 a t - 0)}^{2}} = a (t^{2} + 1) = b$ (given) ...(1)
$S Q = \sqrt{{(\frac{a}{t^{2}} - a)}^{2} + {(- \frac{2 a}{t} - 0)}^{2}} = a (\frac{1}{t^{2}} + 1) = k$ (given) ...(2)
Dividing (1) by (2), we get $t^{2} = \frac{b}{k}$
Putting in (1), $\frac{b}{k} + 1 = \frac{b}{a} \Rightarrow k = \frac{a b}{b - a}$ .

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If b,k are intercepts of a focal chord of the parabola y2=4ax then k is equal to

If $b, k$ are intercepts of a focal chord of the parabola $y^{2} = 4 a x$ then $k$ is equal to