Question

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 is

• 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

Answer 1

Answer: (A)

The equation of the tangent at the point
$R(x,f(x))$ is $Y-f(x)=f^{\prime }(x)(X-x)$

The coordinates of the point $P$ are $\left(0,f(x)-x{f}^{\prime }(x)\right)$
The slope of the perpendicular line
through $P$ is $\frac{f(x)-x{f}^{\prime }(x)}{-1}=-\frac{1}{f^{\prime }x}$
$\Rightarrow f(x)f^{\prime }(x)-x{\left(f^{\prime }(x)\right)}^2=1$
$\Rightarrow \frac{ydy}{dx}-x{\left(\frac{dy}{dx}\right)}^2=1$
which is required differential equation to the curve at $y=f(x)$