Circles
Circumcircle of triangle with chord
MJAT_TS5_P1
Grade 12
Question:
Let $O$ be the centre of the circle $x^2+y^2=r^2$, where $r>\dfrac{5}{2}$. Suppose $PQ$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2x+4y=5$. If the centre of the circumcircle of $\triangle OPQ$ lies on the line $x+2y=4$, then the value of $r$ is:
Step-by-Step Solution
Key Concept: The circumcircle of $\triangle OPQ$ passes through $O(0,0)$, $P$, $Q$. Let its equation be $x^2+y^2+Dx+Ey=0$ (since $O$ is on it). The chord $PQ$ is the radical axis: $Dx+Ey+r^2=0$, which must coincide with $2x+4y-5=0$. So $D/2=E/4=-r^2/(-5)=r^2/5$, giving $D=2r^2/5$, $E=4r^2/5$.
$r=\mathbf{3}$.
Correct Answer: A