The sum of the lengths of the sub-tangent and tangent drawn at the point $(x, y)$ on the curve $y = a \log (x^{2} - a^{2})$ varies as

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$x y$

$y / x$

$y^{2}$

$x^{2}$

answer is C.

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Detailed Solution

$Δ = \frac{1}{2} x_{1}^{2} | \frac{m^{2} + 1}{m} |$

$\frac{d y}{d x} = a \cdot \frac{2 x}{x^{2} - a^{2}} = \frac{2 a x}{x^{2} - a^{2}}$

length of subtangent $\begin{aligned} = y \frac{d x}{d y} \\ = y \cdot \frac{(x^{2} - a^{2})}{2 a x} \end{aligned}$

Length of tangent $\begin{aligned} = y \sqrt{1 + {(\frac{d x}{d y})}^{2}} \\ = y \sqrt{1 + {(\frac{x^{2} - a^{2}}{2 a x})}^{2}} \\ = y \sqrt{\frac{4 a^{2} x^{2} + (x^{2} - a^{2})}{4 a^{2} x^{2}}} \\ = y \cdot \frac{y}{2 a x} \sqrt{{(x^{2} + a^{2})}^{2}} = y \frac{(x^{2} + a^{2})}{2 a x} \end{aligned}$

$\begin{aligned} = y \frac{(x^{2} - a^{2})}{2 a x} + y \frac{(x^{2} - a^{2}}{2 a x} \\ = \frac{y}{2 a x} (x^{2} - a^{2} + x^{2} - a^{2}) \\ = \frac{x y}{a} = \frac{1}{a} (x y) \end{aligned}$