Parabola
Consider the curve traced by the parametric pair x = 2t^2 and y = 4t as the parameter t varies over all real numbers; identify the Cartesian equation.
Select the correct option:
Solution
y2=8x
Eliminating the parameter from a parametric pair recovers the Cartesian relation, and recognizing the resulting form lets us classify the conic. From y = 4t we get t = y/4, and substituting into x = 2t^2 gives x = 2(y/4)^2 = 2 · y^2/16 = y^2/8. Multiplying through, y^2 = 8x, a right-opening parabola. Option y^2 = 4x would arise from x = t^2 rather than 2t^2. Option x^2 = 8y wrongly treats x as the squared parameter expression. Option y^2 = 16x comes from mis-squaring y = 4t as if x = t^2. This is the standard JEE Advanced parametric-to-Cartesian elimination, a recurring conic technique. Plausibility check: comparing y^2 = 8x with y^2 = 4ax gives a = 2, and sample t = 1 yields the point (2, 4), which satisfies 16 = 8·2, confirming the curve passes through the generated points consistently.
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About This Question
- Subject
- mathematics
- Chapter
- coordinate geometry
- Topic
- parabola
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
y2=8x
Eliminating the parameter from a parametric pair recovers the Cartesian relation, and recognizing the resulting form lets us classify the conic. From y = 4t we get t = y/4, and substituting into x = 2t^2 gives x = 2(y/4)^2 = 2 · y^2/16 = y^2/8. Multiplying through, y^2 = 8x, a right-opening parabola. Option y^2 = 4x would arise from x = t^2 rather than 2t^2. Option x^2 = 8y wrongly treats x as the squared parameter expression. Option y^2 = 16x comes from mis-squaring y = 4t as if x = t^2. This is the standard JEE Advanced parametric-to-Cartesian elimination, a recurring conic technique. Plausibility check: comparing y^2 = 8x with y^2 = 4ax gives a = 2, and sample t = 1 yields the point (2, 4), which satisfies 16 = 8·2, confirming the curve passes through the generated points consistently.
This medium difficulty mathematics question is from the chapter coordinate geometry, covering the topic of parabola. It appeared in the 2025 exam.
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