A curve $C$ has to the property that if the tangent drawn at any point $P$ on $C$ meets the coordina

Question

A curve $C$ has to the property that if the tangent drawn at any point $P$ on $C$ meets the coordinate axes at $A$ and $B$, then $P$ is the mid-point of $A B$. The curve passes through the point $(1,1)$. The equation of the curve is

Professor PC Thomas Classes · Answer

Equation of the tangent at point\ ( $x, y$ ) on the curve is\ $Y-y=d y / d x(X-x)$\ This meets axes in $A(x-y d x / d y, 0)$ and $B(0$, $y-x d y / d x)$ mid point of $A B$ is\ $\left[\frac{1}{2}\left(x-y \frac{d x}{d y}ight), \frac{1}{2}\left(y-x \frac{d y}{d x}ight)ight]$\ Given $\frac{1}{2}\left(x-y \frac{d x}{d y}ight)=x$\ and $\frac{1}{2}\left(y-x \frac{d y}{d x}ight)=y$\ $\frac{x d y}{d x}=-y$ or $\frac{d y}{y}=-\frac{d x}{x}$\ Integrating, we get $\log y=-\log x+a$\ Putting $x=1, y=1$, we get\ $\log 1=-\log 1+a \Rightarrow a=0$\ $	herefore \quad \log y+\log x=0 \Rightarrow \log (x y)=0$\ $\Rightarrow x y=e^{0}=1$\ which is a rectangular hyperbola.\