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Which Parametric Equation is Not a Parabola? | JEE

JEE Maths question with a full step-by-step solution.

Question
Which of the following parametric equations does not represent a parabola?
Ax=t2+2t+1, y=2t+2x = t^2 + 2t + 1,\ y = 2t + 2
Bx=a(t2−2t+1), y=2at−2ax = a(t^2 - 2t + 1),\ y = 2at - 2a
Cx=3sin⁡2t, y=6sin⁡tx = 3\sin^2 t,\ y = 6\sin t
Dx=asin⁡t, y=2acos⁡tx = a\sin t,\ y = 2a\cos tcorrect
Solution
Step 1: The first three reduce to parabolas.
(1) x=(t+1)2, y=2(t+1)⇒y2=4x;(2) x=a(t−1)2, y=2a(t−1)⇒y2=4ax;(1)\ x = (t+1)^2,\ y = 2(t+1) \Rightarrow y^2 = 4x; \qquad (2)\ x = a(t-1)^2,\ y = 2a(t-1) \Rightarrow y^2 = 4ax;
(3) y2=36sin⁡2t=12(3sin⁡2t)=12x.(3)\ y^2 = 36\sin^2 t = 12(3\sin^2 t) = 12x.
 Step 2: For the last,
(xa)2+(y2a)2=sin⁡2t+cos⁡2t=1,\left(\frac{x}{a}\right)^2 + \left(\frac{y}{2a}\right)^2 = \sin^2 t + \cos^2 t = 1,
 an ellipse — not a parabola. Correct answer: (4)
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