If a tangent to the circle x^2+y^2=1 intersects the coordinate axes at distinct points P a…

Question

If a tangent to the circle $x^2+y^2=1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the midpoint of $PQ$ is:

Accepted Answer

Tangent: $x\cos\theta+y\sin\theta=1$; $P=\left(\dfrac1{\cos\theta},0\right)$, $Q=\left(0,\dfrac1{\sin\theta}\right)$; midpoint $(h,k)=\left(\dfrac1{2\cos\theta},\dfrac1{2\sin\theta}\right)$.

$$\frac1{h^2}+\frac1{k^2}=4\ \Longrightarrow\ h^2+k^2=4h^2k^2.$$

Solutions · 1