Question

The sum of the lengths of tangents and subtangent at a point of $y=a\log (x^2-a^2)$,$(a>0)$ is porportional to

• A. $\left|x\right|$
• B. $\left|y\right|$
• C. $\left|xy\right|$
• D. $\left|x/y\right|$

Answer 1

The correct option is C $\left|xy\right|$

$y=a\log (x^2-a^2).a>0.$

$\frac{dy}{dx}=\frac{2ax}{x^2-a^2}$

Length of tangent $=y\sqrt{1+\frac{1}{{\left(\frac{dy}{dx}\right)}^2}}$

$=\sqrt{1+\frac{(x^2-a^2{)}^2}{(2ax{)}^2}}$

$=\frac{|y(x^2+a^2)|}{|2ax|}$

Length of subtangent $=\left|y\frac{dx}{dy}\right|=|y\times \frac{1}{\frac{dy}{dx}}|$

$=\frac{y(x^2-a^2)}{|2ax|}$

Sum $=\frac{|x^{2}y|}{|2ax|}=\frac{|xy|}{2a}$

Proportional to $|xy|$