Circles
Circumcircle of Triangle Formed by Tangents from Origin
nta_pyq_2023_apr
Grade 11
Question:
Let $O$ be the origin and $OP$ and $OQ$ be the tangents to the circle $x^2+y^2-6x+4y+8=0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $OPQ$ passes through the point $\left(\alpha,\dfrac{1}{2}\right)$, then a value of $\alpha$ is
$\dfrac{3}{2}$
$\dfrac{-1}{2}$
$\dfrac{5}{2}$
$1$
Step-by-Step Solution
Key Concept: Since $OP\perp CP$ and $OQ\perp CQ$ (tangent⊥radius), the circumcircle of $\triangle OPQ$ has $OC$ as diameter, where $C=(3,-2)$ is the circle's centre.
Circumcircle: $x^2+y^2-3x+2y=0$. For $(\alpha,\frac{1}{2})$: $4\alpha^2-12\alpha+5=0\Rightarrow\alpha=\frac{5}{2}$.
Correct Answer: 3