Question

A curve $y=f\left(x\right);(y>0)$ passes thorugh $(1,1)$ and at point $P(x,y)$ tangents cuts x-axis and y-axis at A and B respectively. If P divides AB internally in the ratio $3:2$, then the value of $f\left(\frac{1}{8}\right)$ is

• A. $4$
• B. $\frac{1}{4}$
• C. $16\sqrt{2}$
• D. $\frac{1}{16\sqrt{2}}$

Answer 1

The correct option is A $4$

$\frac{dy}{dx}=-\frac{k}{h}=-\frac{2y}{3x}$
$\Rightarrow 3\frac{dy}{y}+2\frac{dx}{x}=0$
$\Rightarrow 3\ln y^3{x}^3=C$
$\because$passing through $(1,1)$
$\Rightarrow c=0$
$\Rightarrow y^3{x}^2=1$
at $x=\frac{1}{8}$

$\Rightarrow y=4$