question_answer

Tangent to a curve intercepts the 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

A) \[y\frac{dy}{dx}-x{{\left( \frac{dy}{dx} \right)}^{2}}=1\]

B) \[\frac{x{{d}^{2}}y}{d{{x}^{2}}}+{{\left( \frac{dy}{dx} \right)}^{2}}=0\]

C) \[y\frac{dx}{dy}+x=1\]

D) None of these

Correct Answer: A

Solution :

[a] The equation of the tangent at the point \[R(x,f(x))\]is \[Y-f(x)=f'(x)(X-x)\] The coordinates of the point P are\[(0,f(x)-xf'(x))\]. The slope of the perpendicular line through P is \[\frac{f(x)-xf'(x)}{-1}=-\frac{1}{f'(x)}\] Or \[f(x)f'(x)-x{{(f'(x))}^{2}}=1\] Or \[\frac{ydy}{dx}-x{{\left( \frac{dy}{dx} \right)}^{2}}=1\] Which is the required differential equation to the curve at \[y=f(x).\]