Question

Answer the following :

Two circles each of radius 7 cm, intersect each other. The distance between their centres is `7sqrt(2)` cm. Find the area common to both the circles.

Answer 1

Let O and O₁ be the centres of two circles intersecting each other at A and B.

Then OA = OB = O₁A = O₁B = 7 cm and OO₁ = `7sqrt(2)`

OO₁² = 98  ......(i)

Since OA² + O₁A² = 7² + 7² 

= 98

= OO₁²  .....[From (i)]

∴ m∠OAO₁ = 90°

∴ m∠OAO₁B is a square.

m∠AOB = m∠AO₁B = 90°

= `(90 xx pi/180)^"c" = (pi/2)^"c"`

Now, A(sector OAB) = `1/2"r"^2theta`

= `1/2 xx 7^2 xx pi/2`

= `(49pi)/4"sq.cm"`

And A(sector O₁AB) = `1/2"r"^2theta`

= `1/2 xx 7^2 xx pi/2`

= `(49pi)/4"sq.cm"`

A(∠OAO₁B) = (side)² = (7)² = 49 sq.cm

∴ Required area = area of shaded portion = A(sector OAB) + A(sector O₁AB) – A(∠OAO₁B)

= `(49pi)/4 + (49pi)/4 - 49`

= `(49pi)/2 - 49`

= `49(pi/2 - 1)"sq.cm"`