Question

Find the area of the region common to the circle x² + y² = 16 and the parabola y² = 6x.

Answer 1

There are two equations,

And when x = 2 then y = ±2√3

Equation (i) represents a circle with centre (0, 0) and radius 4 units, so it meets x - axis at (±4, 0) and equation (ii) represents a parabola with vertex (0,) and axis as x - axis

Points of intersection of parabola and circle are (2, 2√3) and (2, – 2√3).

The sketch of the two curves are drawn below,

The shaded region represents the required area.

Required area = Region OBCAO

Required area = 2 (region ODAO + region DCAD)

So, the required area is 2\(\big\{\frac{2\sqrt{3}}{3} + \frac{8\pi}{3}\big\}\) sq units.