The correct option is C an ellipse
Given, x=2(cost+sint)
⇒x2=(cost+sint) ⋯(i)
and y=5(cost−sint)
⇒y5=(cost−sint) ⋯(ii)
From (i) and (ii),
(x2)2+(y5)2
=(cost+sint)2+(cost−sint)2
=cos2t+sin2t+2costsint+cos2t+sin2t−2costsint
=2(cos2t+sin2t)
=2×1=2
⇒x24+y225=2
⇒25x2+4y2−200=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+2fgh−af2−bg2−ch2
=25×4×(−200)+0−0−0−0
=−20000≠0 ⋯(iii)
Now, h2−ab=0−100=−100<0 ⋯(iv)
From (iii) and (iv), the above equation represents an ellipse.