Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $C$ be a circle with centre 'O' and $HK$ is the chord of contact of tangents drawn from a point $A$. $OA$ intersects the circle 'C' at $P$ and $Q$ and $B$ is the midpoint of $HK$, then:
$AB$ is the harmonic mean of $AP$ and $AQ$
$OA$ is the arithmetic mean of $AP$ and $AQ$
$(AK)^2 = (OA)(AB)$
$AB$ is the geometric mean of $AP$ and $AQ$
Step-by-Step Solution
Key Concept: The harmonic mean relationship emerges from equating two expressions derived from the angle bisector and power of a point.
Using the angle bisector property and midpoint theorem, $\frac{AP + AQ}{2} = OA$ where $O$ is the center. From $\cos\theta = \frac{AK}{OA} = \frac{AB}{AK}$, we get $(AK)^2 = (OA)(AB)$. Combining these results yields $AB = \frac{2(AP)(AQ)}{AP + AQ}$, which is the harmonic mean formula.
Correct Answer: 1,2,3