Question

If a variable tangent to the curve $x^{2}y=c^{}$ makes intercepts $a,b$ on $x$ and $y-$axis respectively, then the value of $a^{2}b$ is

• A. $27c$
• B. $\frac{4}{27}c$
• C. $\frac{27}{4}c$
• D. $\frac{4}{9}c$

Answer 1

Answer: (C)

$\frac{27}{4}c$

Given

Curve $x^{2}y=c$

find $a^{2}b=?$

Let the point be $(h,k)$ in the curve $x^y=c$ where the variable tangent is drawn,

then, the equation of tangent $=$ differentiate the curve equation

$\Rightarrow x^2\frac{dy}{dx}+2xy=0$

$\Rightarrow \frac{dy}{dx}=-\frac{2y}{x}$ (slope of tangent)

$\to$ equation of tangent at point $(h,k)$

$\Rightarrow y-y_1=$ slope $(x,x_1)$

$\Rightarrow y-k=-\frac{2k}{h}(x-h)---\left(2\right)$

Now, Given $x-$intercept $=a$

$y-$intercept $=b$

to find $x$ intercept, point $y=0$ in equation $\left(1\right)$

then, $x=\frac{3h}{2}=a$

it $x=0,\Rightarrow y=3k=b$

then, $a^{2}b=\frac{27}{4}{h}^{2}k$

at, $(h,k)$ lies on curve so

$a^{2}b=\frac{27}{4}c$