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MathsMediumClass 12
Asymptotes, eccentricity
Conic Sections
8
JEE Qs
8%
Hard
60
min
Master the formulas and conceptual understanding of eccentricity for all conics, and practice deriving and applying asymptote equations specifically for hyperbolas in various forms.
🧮 Key Formulas
Eccentricity (e) definition: e = c/a (where c is distance from center to focus, a is distance from center to vertex)
Relationship for Ellipse: c^2 = a^2 - b^2 OR b^2 = a^2(1 - e^2) => e = sqrt(1 - b^2/a^2) (if a > b)
Relationship for Hyperbola: c^2 = a^2 + b^2 OR b^2 = a^2(e^2 - 1) => e = sqrt(1 + b^2/a^2)
Asymptotes of Hyperbola (x^2/a^2 - y^2/b^2 = 1): y = ±(b/a)x
Asymptotes of Hyperbola (y^2/b^2 - x^2/a^2 = 1): x = ±(b/a)y
Combined equation of asymptotes for x^2/a^2 - y^2/b^2 = 1: x^2/a^2 - y^2/b^2 = 0
Angle (2θ) between asymptotes: tan(θ) = b/a, so 2θ = 2 * tan⁻¹(b/a)
✅ Key Points for JEE
- 1Eccentricity (e) is a dimensionless parameter that defines the shape of a conic section; e = 0 for circle, 0 < e < 1 for ellipse, e = 1 for parabola, e > 1 for hyperbola.
- 2Asymptotes are lines that a hyperbola approaches but never touches as its branches extend infinitely; they exist only for hyperbolas.
- 3The equation of the asymptotes of a hyperbola x²/a² - y²/b² = 1 is obtained by replacing the '1' on the RHS with '0' (i.e., x²/a² - y²/b² = 0).
- 4For a rectangular hyperbola (a = b), the asymptotes are perpendicular, and its eccentricity e = sqrt(2).
- 5Any line parallel to an asymptote intersects the hyperbola at exactly one point.
⚠️ Common Mistakes
- ✕Confusing the relationship between a, b, and e for ellipses (b² = a²(1-e²)) with that for hyperbolas (b² = a²(e²-1)).
- ✕Incorrectly identifying 'a' and 'b' values, especially when the transverse axis is along the y-axis (y²/b² - x²/a² = 1).
- ✕Attempting to find asymptotes for ellipses or parabolas, as they do not have them.
📝 Practice Questions
See allQ70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1
2019·MCQHard
Q69.If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P , then the distance of P from the origin (units), is: + (1) 2(3 2√2) (2) 3 + 2√2 + (3) √2 + 1 (4) 2(√2 1) JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper
2017·Multi conceptHard
NCERT Chapters
- Class 11 Maths Ch 11: Conic Sections