Question

two circles of radii 5cm and 3cm intersect at point two points and distance between their centres 4 cm find the length of common chord​

Answer 1

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.

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