MathsMediumClass 12

Eccentricity, foci, directrix

Conic Sections

JEE Qs

Hard

min

Master the definition SP = e PM as it is the unifying concept for all conics and provides a robust method for solving problems regardless of conic orientation.

🧮 Key Formulas

Definition of a conic: SP = e PM (distance from point P to focus S = e * distance from P to directrix M)

General equation of conic with focus (x1, y1) and directrix Ax + By + C = 0: (x - x1)^2 + (y - y1)^2 = e^2 * (Ax + By + C)^2 / (A^2 + B^2)

Parabola (y^2 = 4ax): e = 1, Focus (a, 0), Directrix x = -a

Ellipse (x^2/a^2 + y^2/b^2 = 1, a > b): Foci (+-ae, 0), Directrices x = +-a/e, e = sqrt(1 - b^2/a^2), a^2e^2 = a^2 - b^2

Hyperbola (x^2/a^2 - y^2/b^2 = 1): Foci (+-ae, 0), Directrices x = +-a/e, e = sqrt(1 + b^2/a^2), a^2e^2 = a^2 + b^2

✅ Key Points for JEE

1The eccentricity 'e' dictates the type of conic: e=1 for parabola, 0<e<1 for ellipse, e>1 for hyperbola, e=0 for circle (degenerate ellipse).
2The fundamental definition SP = e PM is the most versatile tool; it can be used to derive the equation of any conic given its focus, directrix, and eccentricity, even in non-standard orientations.
3For ellipse and hyperbola, 'ae' represents the distance from the center to a focus, and 'a/e' represents the distance from the center to a directrix.
4Remember the key relations: b^2 = a^2(1-e^2) for ellipse and b^2 = a^2(e^2-1) for hyperbola. These are frequently used to find 'e' or other parameters.
5Be careful with conics whose major/transverse axis is along the y-axis (e.g., x^2/b^2 + y^2/a^2 = 1 for ellipse with a > b); the roles of 'a' and 'b' (and thus 'ae', 'a/e') will swap accordingly with x and y coordinates.

⚠️ Common Mistakes

✕Mixing up the 'a' and 'b' values for ellipses or hyperbolas, especially when the major/transverse axis is along the y-axis.
✕Confusing the formulas for eccentricity (e^2 = 1 - b^2/a^2 vs e^2 = 1 + b^2/a^2) between ellipse and hyperbola.
✕Incorrectly applying 'ae' for directrix and 'a/e' for focus, or vice versa. Focus distance is ae, directrix distance is a/e from the center.
✕Forgetting the general equation from SP=ePM, which is crucial for problems involving non-standard axes or foci.

📝 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 Mathematics Ch 11: Conic Sections