Two circles of radii 5cm and 3cm intersect at two points and the distance between their centres is 4cm. Find the length of common chord.
Let O and O’ be the centres of the circles of radii 5 cm and 3 cm respectively and let PQ be their common chord.
OP = 5 cm, O’P = 3 cm, OO’ = 4 cm.
Let OL = x, so LO’ = 4 - x
Let PQ = y, so PL = y/2
In right triangle OLP,
OL2 = (OP2 – LP2) = 52 – (y/2)2
⇒ x2 = 25 – y2/4 …(i)
In right triangle O’LP, O’L2 = (32 – y2/4)
(4 – x)2 = 9 – y2/4 …(ii)
From (i) and (ii) we get
x2 – 16 + 8x – x2 = 25 – y2/4 – 9 + y2/4
⇒ 8x =32
⇒ x = 4
From equation (i)
16 = 25 – y2/4 => y2/4 = 9 => y2= 36 => y = 6
Thus, the length of the common chord is 6 cm.