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Question

The curve described parametrically by x=2(cost+sint),y=5(costsint) represents

A
a circle
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B
a parabola
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C
an ellipse
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D
a hyperbola
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Solution

The correct option is C an ellipse
Given, x=2(cost+sint)
x2=(cost+sint) (i)
and y=5(costsint)
y5=(costsint) (ii)

From (i) and (ii),
(x2)2+(y5)2
=(cost+sint)2+(costsint)2
=cos2t+sin2t+2costsint+cos2t+sin2t2costsint
=2(cos2t+sin2t)
=2×1=2
x24+y225=2
25x2+4y2200=0

On comparing with ax2+by2+2hxy+2gx+2fy+c=0, we have
a=25,b=4,h=0,g=0,f=0,c=200
Then Δ=abc+2fghaf2bg2ch2
=25×4×(200)+0000
=200000 (iii)
Now, h2ab=0100=100<0 (iv)
From (iii) and (iv), the above equation represents an ellipse.

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