​ The figure given below contains a straight line L with slope √8 and a circle. ​

Question

The figure given below contains a straight line L with slope \(\sqrt{8}\) and a circle.

1. Find the equation of the line L and circle. 
2. Find the point of intersection P.
3. Find the area of the shaded region.

Answer 1

1. The line L passes through origin and have slope 3, therefore its equation is y = \(\sqrt{8}\) x. The circle passes through origin and have radius 3, therefore its equation is x² + y² = 9.

2. We have, y =3x and x² + y² = 9
⇒ x² + (\(\sqrt{8}\)x)² = 9 ⇒ 9x² = 9
⇒ x = 1
∴ y = \(\sqrt{8}\) × 1 = \(\sqrt{8}\).
Therefore, coordinate of ‘P’ is (1, \(\sqrt{8}\)).

3. Area of the shaded region