Find the area common to the circle x^{2} + y^{2} = 16 a^{2} and the parabola y^{2} = 6ax.

OR

Find the area of the region {(x,y):y^{2} ≤ 6ax} and {(x,y):x^{2} + y^{2} ≤ 16a^{2}}.

To find area given equations are

y^{2} = 6ax ...(i)

x^{2} + y^{2} = 16 a^{2} ...(ii)

On solving Equation (i) and (ii)

Or x^{2} + (6ax)^{2} = 16a^{2}

Or x^{2} + (6ax)^{2} – 16a^{2} = 0

Or (x + 8a) (x – 2a) = 0

Or x = 2a or x = – 8a is not possible solution.

Then y^{2} = 6a(2a) = 12a^{2} = 2√3a

Equation (i) represents a parabola with vertex (0,0) and axis as x - axis.

Equation (ii) represents a with centre (0,0) and meets axes (±4a,0), (0,±4a).

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

These are shown in the graph below: -

Required area = 2[Region ODCO + Region BCDB]

The area common to the circlex^{2} + y^{2} = 16 a^{2} and the parabola y^{2} = 6ax is

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