Question

$A,B,C$ are three points on a circle such that $AB$ is the chord and $CP$ is the 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. $r^2={OP}^2+AP\times CP$
• B. $r^2={OP}^2+AP\times PB$
• C. $r^2={OP}^2+PB\times PC$
• D. $r^2={OP}^2+{PB}^2$

Answer 1

Answer: (B)

$r^2={OP}^2+AP\times PB$

In $△ONP$

$O{P}^2=O{N}^2+N{P}^2\to \left(1\right)$

In $△ONA$

$r^2=O{N}^2+N{A}^2$

Putting in $\left(1\right))$

$\Rightarrow O{P}^2=r^2-N{A}^2+N{P}^2\to \left(a\right)$

$NA=NB(\because OAN\cong ONB)$

$\Rightarrow NA-NP=NB-NP=PB\to \left(2\right)$

and $NA+NP=AP\to \left(3\right)$

Solving equation $\left(a\right)$ further:

$\Rightarrow O{P}^2=r^2-(NA-NP)(NA+NP)$

Using $\left(2\right)$ and $\left(3\right)$

$\Rightarrow O{P}^2=r^2-PB\left(AP\right)$

$\Rightarrow r^2=O{P}^2+PA\times PB$

$\therefore B)$ Answer