KCET · Maths · Application of Derivatives
The length of the subtangent to the curve \(x^{2} y^{2}=a^{4}\) at \((-a, a)\) is
- A \(\frac{a}{2}\)
- B \(2 a\)
- C \(a\)
- D \(\frac{a}{3}\)
Answer & Solution
Correct Answer
(C) \(a\)
Step-by-step Solution
Detailed explanation
Given, curve \(x^{2} y^{2}=a^{4}\)
\[
\Rightarrow \quad y^{2}=\frac{a^{4}}{x^{2}}
\]
On differentiating, we get
\[
\begin{aligned}
2 y \frac{d y}{d x} &=\frac{-2 a^{4}}{x^{3}} \\
\Rightarrow & \frac{d y}{d x} &=\frac{-a^{4}}{x^{3} y} \\
\text { at } &(-a, a), \frac{d y}{d x} &=\frac{-a^{4}}{-a^{3} \cdot a}=1
\end{aligned}
\]
Now, length of subtangent to the given curve at \((-a, a)\) is
\[
\frac{y}{\frac{d y}{d x}}=\frac{a}{1}=a
\]
\[
\Rightarrow \quad y^{2}=\frac{a^{4}}{x^{2}}
\]
On differentiating, we get
\[
\begin{aligned}
2 y \frac{d y}{d x} &=\frac{-2 a^{4}}{x^{3}} \\
\Rightarrow & \frac{d y}{d x} &=\frac{-a^{4}}{x^{3} y} \\
\text { at } &(-a, a), \frac{d y}{d x} &=\frac{-a^{4}}{-a^{3} \cdot a}=1
\end{aligned}
\]
Now, length of subtangent to the given curve at \((-a, a)\) is
\[
\frac{y}{\frac{d y}{d x}}=\frac{a}{1}=a
\]
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