Method to find the sum of length of tangent and subtangent

Question

Show that in the curve $y =a\ln(x^2 - a^2)$, sum of the length of tangent & subtangent varies as the product of the coordinates of the point of contact.

Is there any method to solve this type of problem.

Answer 1

We know that $$\text{Length of tangent} = |y|\sqrt{1 + (\frac{dx}{dy})^2} = |y|\sqrt{1 + \frac{1}{(\frac{dy}{dx})^2}}$$ and that $$\text{Length of subtangent} = |y\frac{dx}{dy}| = |\frac{y}{\frac{dy}{dx}}|$$ For the curve, $y = a\ln(x^2-a^2)$, we have, $$\frac{dy}{dx} = \frac{2ax}{x^2-a^2}$$ Thus giving us the length of tangent as $$\text{Length of tangent} = \frac{y(x^2 + a^2)}{2ax}$$ and $$\text{Length of subtangent} =\frac{y(x^2-a^2)}{2ax}$$ Summing up, we get, $$\text{Length of tangent} + \text{Length of subtangent} = \frac{xy}{a} \propto xy$$ Hope it helps.