Question

For the curve $x^2{y}^3=C$ (where $C$ is constant ), the portion of the tangent between the axes is divided by the point of tangents in the ratio of

• A. $3:5$
• B. $2:5$
• C. $3:2$
• D. $3:4$

Answer 1

The correct option is C $3:2$

$\text{We}\text{have}$

$x^2{y}^3=C......\left(1\right)$

$\text{On}\text{differentating}\text{w}\text{.r}\text{.t}\text{.}\text{x}\text{and}\text{we}\text{get}\text{.}$

$\frac{dy}{dx}=\frac{-2y}{3x}$

$\text{Now,}\text{e}quationoftangentatgeneralpoint$

$(x,y)is$

$Y-y=\frac{-2y}{3x}(\times -x)$

$Ifx-intercept=\frac{5}{2}x.$

$y-intercept=\frac{5}{3}y$

$\Rightarrow A=(\frac{5}{2}\times ,0)\text{and}\text{at}AP:PB=k:1$

$\Rightarrow P\equiv (\frac{5x}{2(k+1)},\frac{5ky}{3(k+1)})$

$\Rightarrow \frac{5ky}{3(k+1)}=x\text{and}\frac{5ky}{(3k+1)}=y$

$\Rightarrow k=\frac{3}{2}\text{from}\text{both}\text{Solutions}\text{.}$

$\text{Hence,}\text{the}\text{ratio}\text{is}3:2$

Hence, $\text{this}\text{is}\text{the}\text{answer}\text{.}$