Question

Sketch the region common to the circle x² + y² = 25 and the parabola y² = 8x. Also, find the area of the region, using integration.

Answer 1

Here y² = 8x is a right hand parabola having (0, 0) as the vertex and x² + y² = 25 is a circle having (0, 0) as center and 5 units as radius

x² + y² = 25 ….. (1)

y² = 8x ….. (2)

By solving both the equations we get

x² + 8x – 25 = 0

We get

x = (- 8 ± √(64 + 100)/2

So x = – 4 + √41 is the rejecting negative value

By substituting y = 0 in equation (1) we get x = ± 5

So the circle (1) cuts x-axis at the point C (5, 0) and C’ (-5, 0)

Here the required area = 2 [area of ODCAO]

We can write it as

Required area = 2 [area of ODAO + area of ADCA]