Question

The equation of the curve, such that the intercept cut off by a tangent on the $y-$axis is equal to $n$ times the ordinates of the point of contact of the tangent, is

• A. $y=c{x}^{n-1}$
• B. $y=c{x}^{1-n}$
• C. $y=c{y}^{1-n}$
• D. $y=c{y}^{n-1}$

Answer 1

The correct option is B $y=c{x}^{1-n}$

We have, the equation of tangent to the curve at $(x,y)$ is
$Y-y=\frac{dy}{dx}(X-x)$
$\therefore$ Intercept on $Y-$axis is $y-\frac{xdy}{dx}$
By the given condition,
$y-x\frac{dy}{dx}=ny$
$\Rightarrow \frac{dy}{y}=(1-n)\frac{dx}{x}$
Integrating,
$\log y=(1-n)\log x+\log c$
$\therefore y=c{x}^{1-n}$
Where $c$ is constant of integration