Two circles with centres $O$ and $O'$ of radii $6$ cm and $8$ cm, respectively intersect at two points $P$ and $Q$ such that $OP$ and $O'P$ are tangents to the two circles. Find the length of the comm

Question

Two circles with centres $O$ and $O'$ of radii $6$ cm and $8$ cm, respectively intersect at two points $P$ and $Q$ such that $OP$ and $O'P$ are tangents to the two circles. Find the length of the common chord $PQ$.

Accepted Answer

Step 1: $OP \perp O'P$ (since $OP$ is tangent to the circle with centre $O'$ and $O'P$ is tangent to circle with centre $O$).
$OO' = \sqrt{6^2 + 8^2} = 10$ cm
Step 2: Let $OA = x$, $O'A = 10 - x$ (where $A$ is the intersection of $PQ$ and $OO'$).
Step 3: $AP^2 = 36 - x^2$
Also $AP^2 = 64 - (10 - x)^2$
Step 4: $36 - x^2 = 64 - (10 - x)^2$
$36 - x^2 = 64 - 100 + 20x - x^2$
$36 = -36 + 20x \Rightarrow x = 3.6$
Step 5: In $	riangle PAO$: $AP^2 = 36 - (3.6)^2 = 23.04 \Rightarrow AP = 4.8$
Length $PQ = 2 	imes AP = 9.6$ cm

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