Question

Find the area of the region enclosed between the circles x² + y² = 1 and (x – 1)² + y² = 1.

Answer 1

Consider

x² + y² = 1 …… (1)

(x – 1)² + y² = 1 …… (2)

Here the intersection at the point which is obtained by solving both the equations is

y² = 1 – x²

By substituting it in equation (2) we get

(x – 1)² + 1 – x² = 1

On further simplification

(x – 1)² – x² = 0

We can write it as

(x – 1 – x) (x – 1 + x) = 0

Here we get

– 2x + 1 = 0 where x = ½

Using this in equation (1) we get y = ± √3/2

So both the equations intersect at point A (1/2, √3/2) and B (1/2, –√3/2)

(0, 0) is the center of first circle and radius 1

Similarly (1, 0) is the center of second circle and radius is 1

Here both the circles are symmetrical about x-axis and the required area is shaded here.

So the required area = area OACB = 2 (area OAC)

= 2 [area of OAD + area DCA]