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Question

The area bounded by the curves x=acos3t,y=asin3t is

A
3πa28
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B
3πa216
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C
3πa232
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D
None of the above
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Solution

The correct option is A 3πa28
x=acos3t , y=asin2t
x23+y23=a23
A=2π0xdy
A=a22π0cos3t×3sin2t×costdt
=3a22π0cos2t×sin2tdt (1)
similarily A=2π0ydx
=3a22π0sin2t×cos2tdt (2)
Adding (1) and (2)
$2A=3a^{2}\int_{0}^{2\pi}{\cos^{2}t\times\sin^{2}t}dt
A=3a282π0sin22dt (3)
Similarily A=3a282π0cos2tdt (4)
By putting t=t+π4 in (3)
Adding (3) and (4)
2A=3a282π0dt
A=3a28π

996393_1028220_ans_b75b683b148d4d4790c7831f300d4719.JPG

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