Equation of the curve whose subnormal is constant is given by:

y^{2} = 2 k x + c

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y^{2} = 2 k x^{2} + c

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y^{2} - x^{2} = 2 k x + c

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x^{2} = 2 k y + c

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Solution

The correct option is D $y^{2} = 2 k x + c$
Angle at N $= π - [\frac{π}{2} + π - ψ] = ψ - \frac{π}{2}$
Sub normal, $N K = y c o t (ψ - \frac{π}{2})$
$N K = - y cot (π / 2 - ψ)$
$N K = - y tan ψ$
Given, $N K = C o n s t a n t$
$y tan ψ = - k$
Slope, $tan (π - ψ) = \frac{d y}{d x}$
$tan ψ = - \frac{d y}{d x}$
$- y \frac{d y}{d x} = - k$
$y d y = \int k d x$
On intergrating both sides
$\int (y d y) = \int (k d x)$
$\Rightarrow \frac{y^{2}}{2} = k x + C$
$\Rightarrow y^{2} = 2 k x + C$ is the required equation.

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