If the length of the subnormal of a curve is constant and if it passes through the origin, then the equation of curve is

x^{2} + y^{2} = k^{2}; k ϵ R

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y^{2} = k x; k ϵ R

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x^{2} = k y^{2}; k ϵ R

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x^{2} - y^{2} = k^{2}; k ϵ R

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Solution

The correct option is A $y^{2} = k x; k ϵ R$
Length of the subnormal of the curve is given by $y \frac{d y}{d x}$
Let $k_{1}$ be the constant length of the subnormal
$\Rightarrow y \frac{d y}{d x} = k_{1}$
On integrating on both the sides we get,
$\Rightarrow \int y d y = \int k_{1} d x$
$\Rightarrow \frac{y^{2}}{2} = k_{1} x + c$
As it is passes through origin so, $x = 0, y = 0$
$\Rightarrow c = 0$
$∴ y^{2} = 2 k_{1} x$
here $2 k_{1} = k$
$∴ y^{2} = k x, k ϵ R$ .

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