Question

Prove that area of the triangle formed by any tangent to the curve $xy=c^2$ and coordinate axes is constant.

Answer 1

$xy=c^2\Rightarrow y=\frac{c^2}{x}$

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

Slope of tangent at $(x1,y1)=\frac{-c^2}{x{1}^2}=\frac{-y1}{x1}$

The equation of tangent at $(x1,y1)$ on the wave is

$y-y1=\frac{-y1}{x1}(x-x1)$

or $yx1+y1x=2x1y1=2c^2$

The points intersection is at $(\frac{2c^2}{2y1},\frac{2c^2}{2x1})$ or

at $(x1,y1)$

Area of triangle formed by the portion of the tangent between areas and co-ordinate areas equals.

$\frac{1}{2}\left(\frac{2c^2}{x}\right)\left(\frac{2c^2}{y1}\right)=2c^2=a$ constant.