MathsMediumClass 12

Properties — Focus, directrix, latus rectum

Conic Sections

JEE Qs

Hard

min

Master the general definition PS = ePM as it is the most versatile tool for solving problems related to focus and directrix, especially for non-standard conic forms.

🧮 Key Formulas

General Conic Definition: PS = ePM (distance from point P to focus S equals e times perpendicular distance from P to directrix M)

Eccentricity (e): e = 1 (Parabola), 0 < e < 1 (Ellipse), e > 1 (Hyperbola)

Parabola (y^2 = 4ax): Focus (a, 0), Directrix x + a = 0, Length of Latus Rectum = 4a

Ellipse (x^2/a^2 + y^2/b^2 = 1, a > b): Foci (±c, 0) where c^2 = a^2 - b^2, Directrices x = ±a/e, Length of Latus Rectum = 2b^2/a, Eccentricity e = c/a

Hyperbola (x^2/a^2 - y^2/b^2 = 1): Foci (±c, 0) where c^2 = a^2 + b^2, Directrices x = ±a/e, Length of Latus Rectum = 2b^2/a, Eccentricity e = c/a

✅ Key Points for JEE

1The general definition PS = ePM is the unifying concept for all conics and is crucial for problems involving conics not in standard form or for locus problems.
2Carefully distinguish between the relations c^2 = a^2 - b^2 for an ellipse and c^2 = a^2 + b^2 for a hyperbola; these are often swapped by mistake.
3Correctly identify 'a' and 'b' based on the standard equation and orientation of the major/transverse axis to determine focus and directrix locations.
4Eccentricity 'e' not only defines the type of conic but also quantifies its 'flatness' for an ellipse or 'spread' for a hyperbola.
5Memorize the standard forms and their corresponding properties (focus, directrix, LLR) for parabolas, ellipses, and hyperbolas, as direct application is common.

⚠️ Common Mistakes

✕Confusing the formulas for c^2 (a^2 - b^2 vs. a^2 + b^2) between ellipse and hyperbola.
✕Incorrectly identifying 'a' and 'b' in the equation, especially when the major/transverse axis is along the y-axis or when the equation is not in standard form.
✕Sign errors when writing the equation of the directrix, particularly for parabolas or when the focus is on the negative axis.
✕Forgetting that the general definition PS = ePM applies universally and is useful for non-standard axis orientations or for deriving equations.

📝 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