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If a curve is such that line joining origin to any point $P (x, y)$ on the curve and the line parallel to y-axis through P are equally inclined to tangent to curve at P, then the differential equation of the curve is

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$x {(\frac{d y}{d x})}^{2} - 2 y \frac{d y}{d x} = x$

$x {(\frac{d y}{d x})}^{2} + 2 y \frac{d y}{d x} = x$

$y {(\frac{d y}{d x})}^{2} - 2 x \frac{d y}{d x} = x$

$y {(\frac{d y}{d x})}^{2} - 2 y \frac{d y}{d x} = x$

answer is A.

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Detailed Solution

$\tan θ = | \frac{\frac{d y}{d x} - \frac{y}{x}}{1 + \frac{y}{x} . \frac{d y}{d x}} | = | \frac{1 - 0}{0 + \frac{d y}{d x}} | = | \frac{1}{\frac{d y}{d x}} |$

$\Rightarrow {(\frac{d y}{d x})}^{2} - \frac{2 y}{x} \frac{d y}{d x} = 1$

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