question_answer

DIRECTION: Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct answer (ONLY ONE option is correct) from the following-

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 \[A\] because

Statement-2: \[AK\] is the Geometric mean of \[AB\] and\[AO,\,\,\,OA\] is the arithmetic mean of \[AP\] and \[A\]

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

B)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

C)  Statement-1 is true, Statement-2 is true; Statement-2 is not a 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) \[[A{{K}^{2}}=AP.AQ\]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)) \[\Rightarrow \]            \[AB=\frac{2(AP)(AQ)}{(AP+AQ)}\]