Circles
Tangent and Chord of Contact
Grade 11
Question:
<p>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. Prove or disprove: AB is the Harmonic Mean of AP and AQ.</p>
<p>(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>(C) Statement-1 is true, Statement-2 is false</p>
<p>(D) Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: The midpoint B of chord of contact HK satisfies harmonic mean relation with P and Q because of the symmetric properties of tangents and the perpendicularity of OA to HK.
<p><strong>Analysis:</strong> For a point A outside circle C with tangent points H and K, the chord HK is perpendicular to OA at B. Since B is the midpoint of HK and tangents from A are equal, we have \(AH = AK\). By the property of tangents: \(AH^2 = AP \cdot AQ\) (power of point). Also, \(\frac{2}{AB} = \frac{1}{AP} + \frac{1}{AQ}\) follows from the harmonic mean property. Statement-2 correctly identifies the relationships but the harmonic mean property comes from geometric configuration rather than Statement-2 being its direct cause.</p>
Correct Answer: B