If the tangents at the points P and Q on the circle x^2+y^2-2x+y=5 meet at the point R(94,…

Question

If the tangents at the points $P$ and $Q$ on the circle $x^2+y^2-2x+y=5$ meet at the point $R\left(\dfrac94,2\right)$, then the area of triangle $PQR$ is:

Accepted Answer

Chord of contact $PQ$ from $R$: $xT=0$ gives $\dfrac94x+2y-x+\dfrac94+\dfrac12(y+2)-5=0\Rightarrow x+2y=5$.

Centre $(1,-\tfrac12)$, radius$^2=r^2=1+\tfrac14+5=\tfrac{25}4$. Distance from centre to $PQ$: $p=\dfrac{|1-1-5|}{\sqrt5}=\sqrt5$.

Half-chord length $=\sqrt{r^2-p^2}=\sqrt{25/4-5}=\sqrt5/2$, so $PQ=\sqrt5$.

Distance from $R$ to line $PQ$: $R'=\dfrac{|9/4+4-5|}{\sqrt5}=\dfrac{5/4}{\sqrt5}=\dfrac{\sqrt5}4$.

$$\text{Area}=\frac12\cdot PQ\cdot R'=\frac12\cdot\sqrt5\cdot\frac{\sqrt5}4=\frac58.$$

Solutions · 1