question_answer

A, B, C are three points on a circle such that AB is the chord and CP is perpendicular to OP, where O is the centre and P is any point on AB. The radius r of the circle is given by

A) \[{{r}^{2}}=O{{P}^{2}}+AP\times CP\]

B) \[{{r}^{2}}=O{{P}^{2}}+AP\times PB\]

C) \[{{r}^{2}}=O{{P}^{2}}+PB\times PC\]

D) \[{{r}^{2}}=O{{P}^{2}}+P{{B}^{2}}\]

Correct Answer: B

Solution :

 From right angled \[\Delta \,OPC,\]                                 \[O{{C}^{2}}=C{{P}^{2}}+O{{P}^{2}}\]                 or            \[{{r}^{2}}=C{{P}^{2}}+O{{P}^{2}}\]                         ?..(i)                 Also,      \[CP\times PD=AP\times PB\]                 or            \[C{{P}^{2}}=AP\times PB\] (since \[CP=PD\]                 From equation (i)                                 \[{{r}^{2}}=O{{P}^{2}}+AP\times PB\]