Q. 85.0( 1 Vote )

Find the area common to the circle x2 + y2 = 16 a2 and the parabola y2 = 6ax.
OR

Find the area of the region {(x,y):y2 6ax} and {(x,y):x2 + y2 16a2}.  [CBSE 2013]

Answer :

To find area given equations are

y2 = 6ax ...(i)


x2 + y2 = 16 a2 ...(ii)


On solving Equation (i) and (ii)


Or x2 + (6ax)2 = 16a2


Or x2 + (6ax)2 – 16a2 = 0


Or (x + 8a) (x – 2a) = 0


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


Then y2 = 6a(2a) = 12a2 = 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 circlex2 + y2 = 16 a2 and the parabola y2 = 6ax is

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