MathsMediumClass 12

Standard Equation — x²/a² - y²/b² = 1

Conic Sections

JEE Qs

Hard

min

Master the definitions and relations for 'a', 'b', and 'e', as many problems involve interchanging these to find properties or form equations in varying contexts.

🧮 Key Formulas

x²/a² - y²/b² = 1 (Standard Equation)

b² = a²(e² - 1)

e = sqrt(1 + b²/a²) (Eccentricity)

Foci: (±ae, 0)

Vertices: (±a, 0)

Directrices: x = ±a/e

Length of Latus Rectum: 2b²/a

Equations of Asymptotes: y = ±(b/a)x

Tangent at (x1, y1): xx1/a² - yy1/b² = 1

Condition for line y = mx + c to be tangent: c² = a²m² - b²

Parametric form: (a secθ, b tanθ)

✅ Key Points for JEE

1The fundamental relation b² = a²(e² - 1) is essential for deriving all properties (foci, directrices, latus rectum) from the values of 'a' and 'e'.
2Asymptotes (y = ±(b/a)x) are critical lines that the hyperbola branches approach infinitely, guiding its shape and aiding in sketching.
3For this specific form, the transverse axis lies along the x-axis with length 2a, and the conjugate axis lies along the y-axis with length 2b.
4The eccentricity 'e' is always greater than 1 for a hyperbola, which is a defining characteristic distinguishing it from an ellipse (e < 1) and parabola (e = 1).
5The geometric definition of a hyperbola is the locus of a point whose difference of distances from two fixed points (foci) is a constant (2a).

⚠️ Common Mistakes

✕Confusing the eccentricity formula for hyperbola (e = sqrt(1 + b²/a²)) with that of an ellipse (e = sqrt(1 - b²/a²)).
✕Incorrectly identifying 'a' and 'b' from the equation, especially if the equation is given in a non-standard form or if the other standard form (y²/a² - x²/b² = 1) is mixed up.
✕Errors in applying the tangency condition (c² = a²m² - b²) due to incorrect signs or confusing it with the ellipse condition.
✕Mixing up the coordinates of foci (±ae, 0) and vertices (±a, 0) or their respective signs.

📝 Practice Questions

See all

Q70.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