Take timed practice tests on Ellipse and Hyperbola for JEE Main and JEE Advanced with session-wise drills, score review, and explanation-led revision.
Six 20-question timed sessions plus one 60-question module test. Each question is original and calibrated from the uploaded material pattern without copying PDF wording.
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1. The standard equation of an ellipse with semi-major axis a (along x) and semi-minor axis b is:
Explanation: Standard ellipse: x²/a² + y²/b² = 1 with a > b > 0. c² = a² − b² (c = focal distance). Eccentricity e = c/a (0
2. The eccentricity of a circle is:
Explanation: For a circle, a = b → c = 0 → e = c/a = 0. For ellipse: 0 1.
3. The standard equation of a hyperbola with transverse axis along x is:
Explanation: Standard hyperbola: x²/a² − y²/b² = 1. c² = a² + b². Eccentricity e = c/a > 1. Foci at (±c, 0). Asymptotes: y = ±(b/a)x.
4. For a rectangular hyperbola, the asymptotes are:
Explanation: A rectangular hyperbola has a = b → asymptotes y = ±x (perpendicular). Equivalently, in rotated form: xy = c². Points on this hyperbola: (ct, c/t).
5. The sum of distances from any point on an ellipse to its two foci equals:
Explanation: Defining property of ellipse: for any point P on the ellipse, PF₁ + PF₂ = 2a. This is used in drawing ellipses with string and pins.
6. The length of the latus rectum of x²/a² + y²/b² = 1 (a > b) is:
Explanation: Latus rectum of ellipse = chord through focus perpendicular to major axis. Length = 2b²/a. Similarly for hyperbola x²/a²−y²/b²=1: LR = 2b²/a.
7. The tangent at (a cos θ, b sin θ) to the ellipse x²/a² + y²/b² = 1 is:
Explanation: Using T = 0 formula: tangent at (x₁, y₁) to ellipse: xx₁/a² + yy₁/b² = 1. At parametric point (a cos θ, b sin θ): x(a cos θ)/a² + y(b sin θ)/b² = 1 → (x cos θ)/a + (y sin θ)/b = 1.
8. The director circle of x²/a² + y²/b² = 1 has equation:
Explanation: The director circle of an ellipse is the locus of points from which the two tangents to the ellipse are perpendicular. Its equation is x² + y² = a² + b².
9. For a hyperbola x²/9 − y²/16 = 1, the asymptotes are:
Explanation: Asymptotes of x²/a²−y²/b²=1: y = ±(b/a)x. Here a²=9 (a=3), b²=16 (b=4). Asymptotes: y = ±(4/3)x.
10. The foci of x²/25 + y²/9 = 1 are at:
Explanation: a² = 25, b² = 9 → c² = 25 − 9 = 16 → c = 4. Foci at (±4, 0).
11. The eccentricity of x²/16 − y²/9 = 1 is:
Explanation: a² = 16, b² = 9 (hyperbola). c² = a² + b² = 25 → c = 5. e = c/a = 5/4 > 1 (confirming hyperbola).
12. The area of the ellipse x²/a² + y²/b² = 1 is:
Explanation: Area of ellipse = πab. For a circle (a = b = r): area = πr². The ellipse with semi-axes a and b is a 'stretched' circle.