Coordinate GeometryHigh Yield (2-3 Qs in JEE)

Ellipse Formulas

Standard ellipse x²/a² + y²/b² = 1, eccentricity, foci, auxiliary circle, director circle, and tangent conditions.

7 Formulas•350+ Questions in Bank•

Concept Diagram

Standard Parabola & Conic Geometry

Standard parabola y² = 4ax with focus S(a,0), directrix x = -a, vertex (0,0), and latus rectum length 4a.

e = \sqrt{1 - \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e} \quad (a > b)

Fundamental eccentricity relation for horizontal ellipse.

Examiner Tip: Latus rectum length = 2b²/a.

SP + S'P = 2a

The sum of distances from any point on the ellipse to both foci is constant and equals the major axis length 2a.

Examiner Tip: Commonly used to identify ellipses from locus equations |z - z₁| + |z - z₂| = 2a.

y = mx \pm \sqrt{a^2 m^2 + b^2}

Condition of tangency: c² = a² m² + b².

Examiner Tip: Point of contact is (∓ a²m / √(a²m² + b²), ± b² / √(a²m² + b²)).

x^2 + y^2 = a^2 + b^2

Locus of point of intersection of two mutually perpendicular tangents to the ellipse.

Examiner Tip: Radius of director circle is √(a² + b²).

P(a\cos\theta, b\sin\theta), \quad \text{Tangent: } \frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1

Parametric eccentric angle θ representation of points and tangent line.

Examiner Tip: Eccentric angle θ is the angle made by corresponding point on the auxiliary circle with the major axis.

y = mx \pm \frac{m(a^2 - b^2)}{\sqrt{a^2 + b^2 m^2}}, \quad ax\sec\theta - by\csc\theta = a^2 - b^2

Equation of normal with slope m, and parametric normal at eccentric angle θ.

Examiner Tip: Maximum 4 normals can be drawn from an arbitrary point (h, k) to an ellipse.

x^2 + y^2 = a^2, \quad p_1 \cdot p_2 = b^2

The product of lengths of perpendiculars drawn from the two foci (±ae, 0) to any tangent is constant and equals b².

Examiner Tip: Feet of perpendiculars drawn from the foci to any tangent always lie on the auxiliary circle x² + y² = a².

Frequently tested misconceptions in Ellipse

•Swapping a and b when b > a (vertical ellipse): eccentricity becomes e = √(1 - a²/b²).
•Sum of focal distances is constant 2a, NOT 2b (for horizontal ellipse).