Circles
Circle
Allen Star Batch
Grade 11
Question:
A straight line through the vertex $P$ of a triangle $PQR$ intersect the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle, then:
$\frac{1}{PS} + \frac{1}{ST} \frac{2}{\sqrt{SQ \times SR}}$
$\frac{1}{PS} + \frac{1}{ST} \frac{4}{QR}$
Step-by-Step Solution
Key Concept: The power of a point theorem combined with AM-GM inequality relates the reciprocals of chord segments.
For two chords $PT$ and $QR$ intersecting at $S$, the power of point $S$ gives $PS \cdot ST = QS \cdot SR$. Using harmonic mean inequality and AM-GM: $\frac{1}{PS} + \frac{1}{ST} > \frac{2}{\sqrt{PS \cdot ST}} = \frac{2}{\sqrt{QS \cdot SR}}$. Similarly, $\frac{QS + SR}{2} > \sqrt{QS \cdot SR}$, leading to $\frac{1}{PS} + \frac{1}{ST} > \frac{4}{QR}$.
Correct Answer: 2,4