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Length of Common Chord

A line is dra...

Question

A line is drawn through a fixed point $P (a, b)$ to cut the circle $x^{2} + y^{2} = r^{2}$ at A and B. Then $P A . P B$ is equal to

(a + b)^{2} - r^{2}

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a^{2} + b^{2} - r^{2}

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(a - b)^{2} + r^{2}

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none of these

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Solution

The correct option is B $a^{2} + b^{2} - r^{2}$
Given point $P (a, b)$ and $A B$ is chord which passes through point $P$
$M$ is the midpoint of $A B$ and
$B M = A M (1)$
in $△ P O M$
$(O M)^{2} = (O P)^{2} - (P M)^{2} (O M)^{2} = (a^{2} + b^{2}) - (P M)^{2} (2)$
In $△ A O M$
$(O M)^{2} = (O A)^{2} - (A M)^{2} (O M)^{2} = r^{2} - (A M)^{2} (3)$
From $(2)$ and $(3)$
$(a^{2} + b^{2}) - (P M)^{2} = r^{2} - (A M)^{2} (a^{2} + b^{2}) - r^{2} = (P M)^{2} - (A M)^{2} (a^{2} + b^{2} - r^{2}) = (P M - A M) (P M + A M)$
From $(1)$
$(a^{2} + b^{2} - r^{2}) = P A \cdot (P M + B M) (a^{2} + b^{2} - r^{2}) = P A \cdot P B$

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