Question

The length of sub-tangent to the curve $x^2{y}^2=a^4$ at the point $(-a,a)$ is

• A. $3a$
• B. $2a$
• C. $a$
• D. $4a$

Answer 1

The correct option is C

$a$

Explanation for correct option

Given equation of the curve is $x^2{y}^2=a^4$

We know that the length of the subtangent is $\left|y·\frac{dx}{dy}\right|$

Now $x^2{y}^2=a^4$

$$\begin{gathered} \Rightarrow x^2=\frac{a^4}{y^2} \\ \Rightarrow 2x\frac{dx}{dy}=a^4\left(\frac{-2}{y^3}\right) \\ \Rightarrow \frac{dx}{dy}=a^4\left(\frac{-1}{x{y}^3}\right) \end{gathered}$$

$\therefore {\left|y·\frac{dx}{dy}\right|}_{\left(-a,a\right)}={\left|y\frac{-a^4}{x{y}^3}\right|}_{\left(-a,a\right)}=\left|a\right|=a$

Hence, the correct option is option C.