For the parabola $y^{2} = 4 a x$ , the ratio of the length of the sub-tangent to the abscissa at any point on the curve is

1 : 1

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2 : 1

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x : y

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x^{2} : y

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Solution

The correct option is C $2 : 1$
Let $P (x_{1}, y_{1})$ be any point on the curve
$y^{2} = 4 a x$
$\Rightarrow y_{1}^{2} = 4 a x_{1}$
$\frac{d y}{d x} = \frac{2 a}{y}$
Slope of tangent at $P (x_{1}, y_{1})$ is $m = \frac{2 a}{y_{1}}$
Length of sub-tangent $= | \frac{y_{1}}{m} | = 2 x_{1}$
Required ratio $= \frac{2 x_{1}}{x_{1}} = 2 : 1$

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