question_answer

Let C be a circle with centre \[O\] and \[HK\] is the chord of contact of pair of the tangents from points A. OA intersects the circle \[C\] at \[P\] and \[Q\] and \[B\] is the midpoint of\[HK\], then

Statement 1: AB is the harmonic mean of AP and AQ.

Statement 2: AK is the Geometric mean of AB and AO, OA is the arithmetic mean of AP and AQ.

A)  Statement-1 is false, Statement-2 is true.

B)  Statement-1 is true.statement-2 is true and statement- is correct explanation for statement-1

C)  Statement-1 is true, statement-2 is true and statement-2 is NOT correct explanation for statement-1

D)  Statement-1 is true, Statement-2 is false.

Correct Answer: B

Solution :

 \[\frac{(AK)}{(OA)}=\cos \theta =\frac{AB}{AK}\] \[\Rightarrow \]\[{{(AK)}^{2}}=(AB)(OA)=(AP)(AQ)\]     ?.(1) \[\Rightarrow \]\[[A{{K}^{2}}=AP.AQ\]using power of point A \[]\] Also, \[OA=\frac{AP+AQ}{2}\] \[[AQ-AO=r=AO-AP\Rightarrow 2AO=AQ+AP]\] \[\Rightarrow \]\[(AP)(AQ)=AB\left( \frac{AP+AQ}{2} \right)\] (from (1)) \[\Rightarrow \]\[AB=\frac{2(AP)(AQ)}{(AP+AQ)}\]