Question

If a variable tangent to the curve $x^{2}y=c^3$ makes intercepts $a,b$ on $X$-axes and $Y$- axes, respectively, then the value of $a^{2}b$ is

• A. $27c^3$
• B. $\frac{4}{27}{c}^3$
• C. $\frac{27}{4}{c}^3$
• D. $\frac{4}{9}{c}^3$

Answer 1

The correct option is C $\frac{27}{4}{c}^3$

Let a variable point$(x_1,y_1)$ on the curve $x^{2}y=c^3$
Then slope $\frac{dy}{dx}=-\frac{2y_1}{x_1}$
So, equation of tangent is
$y-y_1=-\frac{2y_1}{x_1}(x-x_1)$
$\Rightarrow y{x}_1-2x{y}_1=3x_1{y}_1$
Using ${x_1}^2{y}_1=c^3$
Equation of tangent becomes $y{x}_1^2+\frac{2c^{3}x}{x_1}=3c^3$
Therefore, X intercept is $a=\frac{3x_1}{2}$
And Y intercept is $b=\frac{3c^3}{{x_1}^2}$
Now $a^{2}b={\left(\frac{3x_1}{2}\right)}^2\left(\frac{3c^3}{{x_1}^2}\right)=\frac{27c^3}{4}$
Hence, option 'C' is correct.