MathsMediumClass 12

Standard Forms — y²=4ax, all 4 orientations

Conic Sections

JEE Qs

Hard

min

Master the visual representation of each standard form along with its properties; sketching them quickly helps in problem-solving.

🧮 Key Formulas

y^2 = 4ax (a>0): Vertex (0,0), Focus (a,0), Directrix x = -a, Axis y = 0, Latus Rectum Length 4a, Endpoints of Latus Rectum (a, +/-2a)

y^2 = -4ax (a>0): Vertex (0,0), Focus (-a,0), Directrix x = a, Axis y = 0, Latus Rectum Length 4a, Endpoints of Latus Rectum (-a, +/-2a)

x^2 = 4ay (a>0): Vertex (0,0), Focus (0,a), Directrix y = -a, Axis x = 0, Latus Rectum Length 4a, Endpoints of Latus Rectum (+/-2a, a)

x^2 = -4ay (a>0): Vertex (0,0), Focus (0,-a), Directrix y = a, Axis x = 0, Latus Rectum Length 4a, Endpoints of Latus Rectum (+/-2a, -a)

✅ Key Points for JEE

1The sign in the equation (e.g., `4ax` vs. `-4ax`) dictates the orientation of the parabola (right, left, up, or down).
2The parameter 'a' (always taken as positive) represents the distance from the vertex to the focus and from the vertex to the directrix.
3The focus always lies on the axis of symmetry, and the directrix is always perpendicular to the axis of symmetry.
4The vertex is the midpoint of the focus and the foot of the perpendicular from the focus to the directrix.
5The length of the latus rectum is always `|4a|`, which is the length of the chord passing through the focus and perpendicular to the axis.

⚠️ Common Mistakes

✕Confusing the variable (x or y) for the focus coordinates and directrix equation for vertical vs. horizontal parabolas (e.g., mixing x=a for focus with y=a for directrix).
✕Incorrectly applying signs for 'a' in focus and directrix for parabolas opening left or down.
✕Forgetting that 'a' in the standard forms `y^2=4ax`, etc., is typically defined as a positive distance, and the sign in the equation dictates the direction.

📝 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