The sum of the length of tangent and subtangent at a point of $y = a l o g (x^{2} - a^{2}) (a > 0)$ is proportional to?

| x |

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| y |

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| x y |

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∣ ∣ ∣ \frac{x}{y} ∣ ∣ ∣

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Solution

The correct option is A $| x y |$
Length of the tangent = $| y | \sqrt{1 + (\frac{d x}{d y})^{2}}$
Length of the sub-tangent = $∣ ∣ ∣ \frac{y d x}{d y} ∣ ∣ ∣$
$\frac{d y}{d x} = \frac{2 a x}{x^{2} - a^{2}}$
Length of the tangent = $| y | \frac{x^{2} + a^{2}}{2 a x}$
Length of the sub-tangent = $| y | \frac{x^{2} - a^{2}}{2 a x}$
Sum = $\frac{| x y |}{a} \propto | x y |$

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