Circles
Chord of Contact — Area of Triangle
nta_pyq_2023_apr
Grade None
Question:
If the tangents at the points $P$ and $Q$ on the circle $x^2+y^2-2x+y=\dfrac{5}{4}$ meet at the point $R\!\left(\dfrac{9}{4},2\right)$, then the area of the triangle $PQR$ is
\dfrac{5}{4}
\dfrac{13}{8}
\dfrac{5}{8}
\dfrac{13}{4}
Step-by-Step Solution
Key Concept: Tangent length $L=\sqrt{S_1}$ where $S_1$ is obtained by substituting $R$ into the circle equation. Then use the relation $\frac{1}{2}\cdot PQ\cdot h=\frac{1}{2}\cdot L\cdot L$ where $h$ is distance from $R$ to chord $PQ$.
$L=\frac{5}{4}$, $CR=\frac{5\sqrt{5}}{4}$, $PQ=\sqrt{5}$. Area $=\frac{5}{8}$.
Correct Answer: 3