JEE Main Mathematics: Parametric equations of conics Practice Questions
7 free multiple-choice questions on Parametric equations of conics (Analytical Geometry) for JEE Main Mathematics, each with the correct answer and a step-by-step explanation. Attempt a question, then reveal the solution to check your reasoning.
Practice thousands more JEE Main questions in the app →Q1. In the parametric representation of a conic section, the parameter θ is typically related to the angle of the line segment connecting the point to the focus or directrix. This is characteristic of which type of parametric representation?
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Correct answer: D — Parametric form using θ
The use of a parameter like θ in conjunction with trigonometric functions (like cos θ, sin θ) is the standard way to represent conic sections parametrically, linking the coordinates to an angular parameter.
Q2. The parametric equations x = a cos θ and y = b sin θ represent which curve?
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Correct answer: D — Ellipse
These equations define the parametric form of an ellipse centered at the origin with semi-major and semi-minor axes a and b respectively.
Q3. The parametric equation of an ellipse is given by x = 5 cos θ and y = 3 sin θ. What is the Cartesian equation of the ellipse?
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Correct answer: D — x²/25 + y²/9 = 1
From the parametric equations, we have cos θ= x/5 and sin θ= y/3. Using the identity cos²θ+ sin²θ= 1, we get (x/5)² + (y/3)² = 1, which simplifies to x²/25 + y²/9 = 1.
Q4. Which of the following equations represents a hyperbola in parametric form?
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Correct answer: D — x = acosh t, y = asinh t
The standard parametric form for a hyperbola is typically x = acosh t and y = asinh t (or similar hyperbolic functions). The other options represent a circle, a parabola (if x=at², y=bt), and a rotated ellipse (if x=acos t, y=bsin t).
Q5. Consider the parametric equations x = 2sin t and y = 2cos t. The locus of the points (x, y) is:
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Correct answer: A — x² + y² = 4
From the equations, we have (x/2)² + (y/2)² = sin² t + cos² t = 1. This simplifies to x² + y² = 4.
Q6. If the parametric equations are x = a sec t and y = a tan t, the resulting curve is a:
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Correct answer: D — Hyperbola
The relationship is x² = a² sec² t and y² = a² tan² t. Since sec² t - tan² t = 1, we get x²/a² = sec² t and y²/a² = tan² t. Substituting yields x²/a² - y²/a² = 1, which is the equation of a hyperbola.
Q7. If a hyperbola is given by the parametric equations x = a cosh t and y = a sinh t, the Cartesian equation of the hyperbola is:
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Correct answer: A — x²/a² - y²/a² = 1
Since cosh² t - sinh² t = 1, substituting x=a cosh t and y=a sinh t yields x²/a² - y²/a² = 1, which is the standard form of a hyperbola.
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