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Question

The area of the region above the x-axis, included between the parabola y2=ax and the circle x2+y2=2axis

A
a2(π423) square units
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B
2a2(π423) square units
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C
a(π423) square units
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D
a2(π423) square units
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Solution

The correct option is A a2(π423) square units
Solving the given equations of curves, we have
x2+ax=2axx2ax=0x(xa)=0x=0 or awhich gives, y=a or ±a
Area ODAB=a0 (2axx2ax)dxLet x=2a sin2 t. Then dx=4a sin t cos t dtWhen x=0t=0 and,When x=at=π4Again, a0 2axx2 dx=π/40(2a sin t cos t)(4a sin t cos t) dt=a2π/402×(2 sin t cos t)2 dt=a2π/402 sin2 2t dt=a2π/40(1cos 4t) dt=a2(tsin 4t4]π/40=πa24Now, a0 ax dx=a×23(x3/2]a0=23a2Thus, the required area=πa2423a2=a2(π423)

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