# Ellipse Formulas

Ellipse equations calculations can be quite difficult using traditional calculation process. But can be possible to make to easy using Basic Ellipse formulas. With the help of Ellipse Formulae List, you can complete your calculations much easier and faster. So, memorize these basic to advanced ellipse formulas by our provided Ellipse formulas List & Cheatsheet.

## List of Basic & Complex Ellipse Formulas

Want to learn and solve all complex problems on Ellipse? Then, make use of these below-provided ellipse concepts formulae list. Also, remember the formulas by learning daily at once and attempt all ellipse concept easily in the exams.

1. The general equation of an ellipse whose focus is (h, k) and the directrix is the line ax + by + c = 0 and the eccentricity will be e is SP = ePM

General form:

(x_{1} – h)^{2} + (y_{1} – k)^{2} = \(\frac{e^{2}\left(a x_{1}+b y_{1}+c\right)^{2}}{a^{2}+b^{2}}\), e < 1

2. The equation ax^{2} + by^{2} + 2hxy + 2gx + 2fy + c = 0 will represent an ellipse if h^{2} – ab < 0 & ∆ = abc + 2fgh – af^{2} – bg^{2} – ch^{2} ≠ 0.

3. **Standard form of the equation of ellipse** \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1

(A)

- Length of Major axis A
_{1}A_{2}→ 2a - Length of Minor axis B
_{1}B_{2}→ 2b - Directrix: x = \(\frac{a}{e}\) and x = –\(\frac{a}{e}\)
- Focus: S(ae, o) & S'(-ae, 0)
- Length of latus rectum = \(\frac{2 b^{2}}{a}\)

(B)

- Length of Major axis A
_{1}A_{2}= 2b - Length of Minor axis B
_{1}B_{2}= 2a - Directrix: x = \(\frac{b}{e}\),
- Foci: (0, ±be)
- Length of latus rectum = \(\frac{2 a^{2}}{b}\)

4. **Eccentricity b ^{2} = a^{2} (1 – e^{2})** ⇒ e = \(\sqrt{1-\frac{b^{2}}{a^{2}}}\)

5. (a) The equation of ellipse in the parametric form will be given by x = a cos Φ, y = b sin Φ

(b) The position of a point P(x_{1}, y_{1}) with respect to ellipse S = \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) – 1 = 0 can be given in following manner if S_{1} = \(\frac{x_{1}^{2}}{a^{2}}+\frac{y_{1}^{2}}{b^{2}}\) – 1 then

S_{1} < 0 Point is inside

S_{1} = 0 Point lies on ellipse

S_{1} > 0 Point lies out side the ellipse

**6. Condition of tangency**

The line y = mx + c touches the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1, if c = ± \(\sqrt{a^{2} m^{2}+b^{2}}\)

**7. Equation of the Tangent**

(i) The equation of the tangent at any point (x_{1}, y_{1}) on the ellipse

\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{x x_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}\) = 1.

(ii) The equation of tangent at any point ‘Φ’ is \(\frac{x}{a}\) cos Φ + \(\frac{y}{b}\) sin Φ = 1.

(iii) Slope Form: y = mx ± \(\sqrt{a^{2} m^{2}+b^{2}}\) and Its point of contact is

\(\left(\frac{\mp \mathrm{a}^{2} \mathrm{m}}{\sqrt{\mathrm{a}^{2} \mathrm{m}^{2}+\mathrm{b}^{2}}}, \frac{\pm \mathrm{b}^{2}}{\sqrt{\mathrm{a}^{2} \mathrm{m}^{2}+\mathrm{b}^{2}}}\right)\)

**8. Equation of the Normal**

(i) The equation of the normal at any point (x_{1}, y_{1}) on the ellipse

\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{a^{2} x}{x_{1}}-\frac{b^{2} y}{y_{1}}\) = a^{2} – b^{2}

(ii) The equation of the normal at any point ‘Φ’ is ax sec Φ – by cosec Φ = a^{2} – b^{2}

(iii) If eccentric angles of feet P, Q, R, S of these normals be α, β, γ, δ then α + β + γ + δ = (2n + 1)π, n ∈ I

(iv) The necessary and sufficient condition for the normals at three α, β, γ points on the ellipse to be concurrent if

sin(β + γ) + sin(γ + α) + sin(α + β) = 0.

(v) If the normal at any point P on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 meets the axes in G and g respectively then

(a) PG : Pg = b^{2}: a^{2}

(b) PF . PG = b^{2}

(c) PF . Pg = a^{2}, where “CF” is the perpendicular to normal and “C” is centre.

9. The equation of the chord of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1, whose mid point be (x_{1}, y_{1}) is T = S_{1}.

10. The equation of the chord of contact is \(\frac{x x_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}\) = 1 or T = 0 at (x_{1}, y_{1}).

**11. Pair of tangents: SS _{1} = T^{2}**

**12. Pole and Polar**

(i) Equation of polar of P(x_{1}, y_{1}) w.r.t. \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{x x_{1}}{a^{2}}+\frac{y y_{1}}{b^{2}}\) = 1

(ii) The pole of the line lx + my + n = 0 with respect to the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is \(\left(-\frac{\mathrm{a}^{2} \ell}{\mathrm{n}},-\frac{\mathrm{b}^{2} \mathrm{m}}{\mathrm{n}}\right)\)

**13. Director Circle**

The equation of the director circle of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is x^{2} + y^{2} = a^{2} + b^{2}.

**14. Diameter**

If y = mx + c represent a system of parallel chords of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is then the equation of the diameter is y = –\(\frac{b^{2}}{a^{2} m}\)x.

15. If α, β, γ, δ be the eccentric angles of the four concyclic points on an ellipse then α + β + γ + δ = 2nπ.

**16. Reflection property on an ellipse:**

If an incoming light ray passes through one focus strike the concave side of the ellipse then it will get reflected towards other focus.

∠S_{1}PS_{2} = ∠S_{1}QS_{2}