Tangent to a curve intersects y-axis at a point P. A line perpendicular to this tangent through P passes through another point (1, 0). The differential equation of the curve is

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$x \frac{d^{2} y}{{dx}^{2}} + {(\frac{dy}{dx})}^{2} = 0$

$x \frac{dy}{dx} + y = 5$

$y \frac{dx}{dy} + x = 1$

$y \frac{dy}{dx} - x {(\frac{dy}{dx})}^{2} = 1$

answer is A.

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Detailed Solution

Equation of tangent to the curve y=f(x) at (x,y) on it is $Y - y = \frac{dy}{dx} (X - x); P = (0, y - x \frac{dy}{dx})$ ;

$PA$ is perpendicular to tangent at P

$\Rightarrow (x \frac{dy}{dx} - y) (\frac{dy}{dx}) = - 1; y (\frac{dy}{dx}) - x {(\frac{dy}{dx})}^{2} = 1$

Which is differential equation of the curve

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