Question

A curve passing through the point $(1,1)$ is such that the intercept made by a tangent to it on x-axis is three times the x co-ordinate of the point of tangency, then the equation of the curve is:

• A. $y=\frac{1}{x^2}$
• B. $y=\sqrt{x}$
• C. $y=\frac{1}{\sqrt{x}}$
• D. none

Answer 1

The correct option is C $y=\frac{1}{\sqrt{x}}$

Equation of tangent on any point on the curve is,
$Y-y=\frac{dy}{dx}(X-x)$
Thus intercept on the x-axis is,$I_x=x-\frac{y}{dy/dx}$
Now using given condition,
$I_x=x-\frac{y}{dy/dx}=3x$
$\therefore y\frac{dx}{dy}+2x=0$
or $\frac{dx}{2x}+\frac{dy}{y}=0$ or $\frac{1}{2}\log x+\log y=k$ or $\log y\sqrt{x}=e^k=constant$
Now given this curve passing through (1,1) $\Rightarrow k=0$
$\therefore$ Hence required curve is, $y=\frac{1}{\sqrt{x}}$