Question

Show that the area of the triangle formed by the tangent at any point on the curve $xy=c(c\ne 0),$ with the coordinate axes is constant.

Answer 1

$\begin{array}{c}xy=c^2\Rightarrow y=\frac{c^2}{x}\\ \frac{dy}{dx}=\frac{-c^2}{x^2}\\ Slopeofthe\tan gentat\left(x,y\right)=\frac{-c^2}{x^2}=\frac{-y_1}{x_1}\\ Theequationofthe\tan gentat\left(x_1,y_1\right)onthecurveis{y}_1=\frac{-y_1}{x_1}\left(x-x_1\right)\\ or,y{x}_1+y_{1}x=2x_1,y_1=2c^2\\ Thepointsofinter\sec tionofthe\tan gentwiththeaxesare\left(\frac{2c^2}{y_1},0\right)and\left(0,\frac{2c^2}{x}\right)\\ Themidpointofthisinterceptisat\left(\frac{2c^2}{2y_1},\frac{2c^2}{2x_1}\right)orat\left(x_1,y_1\right)\\ Areaofthetriangleformedbytheportionofthe\tan gentbetweentheaxesandthecoordinatesaxesequals\\ \frac{1}{2}\left(\frac{2c^2}{x}\right)\left(\frac{2c^2}{y_1}\right)=2c^2=acons\tan t.\end{array}$