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Observe the following statement for the curve $y^{2} = 4 ax$
Statement-I : The length of the subnormal at any point is a constant
Statement-II : The length of the sub-tangent at any point is twice the abscissa of the point of contact.
Statement-III : Area of triangle formed by tangent, normal and x-axis at any point is a constant.
Out of these, correct statement are

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$I$ and $II$

$II$ and $III$

$I, II, III$

answer is A.

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Detailed Solution

$y^{2} = 4 ax \Rightarrow 2 y \frac{dy}{dx} = 4 a \Rightarrow \frac{dy}{dx} = \frac{2 a}{y}$

$(I) L . S . N = |y_{1} m| = y \cdot \frac{2 a}{y} = 2 a =$ constant

$(II) L . S . T = |\frac{y_{1}}{m}| = \frac{yy}{2 a} = \frac{y^{2}}{2 a} = 2 x$

$(III) x = {at}^{2} y = 2 at$

eq of Tangent $yt = x + {at}^{2}$

eq of normal $xt + y = 2 at + {at}^{3}$

area

$= \frac{1}{2} x (2 a + 2 {at}^{2}) (2 at)$

$\neq$ Constant

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