Ellipse Formulas

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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:
(x₁ – h)² + (y₁ – k)² = \(\frac{e^{2}\left(a x_{1}+b y_{1}+c\right)^{2}}{a^{2}+b^{2}}\), e < 1

2. The equation ax² + by² + 2hxy + 2gx + 2fy + c = 0 will represent an ellipse if h² – ab < 0 & ∆ = abc + 2fgh – af² – bg² – ch² ≠ 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 A₁A₂ → 2a
• Length of Minor axis B₁B₂ → 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 A₁A₂ = 2b
• Length of Minor axis B₁B₂ = 2a
• Directrix: x = \(\frac{b}{e}\),
• Foci: (0, ±be)
• Length of latus rectum = \(\frac{2 a^{2}}{b}\)

4. Eccentricity b² = a² (1 – e²) ⇒ 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(x₁, y₁) with respect to ellipse S = \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) – 1 = 0 can be given in following manner if S₁ = \(\frac{x_{1}^{2}}{a^{2}}+\frac{y_{1}^{2}}{b^{2}}\) – 1 then
S₁ < 0 Point is inside
S₁ = 0 Point lies on ellipse
S₁ > 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 (x₁, y₁) 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 (x₁, y₁) on the ellipse
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\) = 1 is \(\frac{a^{2} x}{x_{1}}-\frac{b^{2} y}{y_{1}}\) = a² – b²

(ii) The equation of the normal at any point ‘Φ’ is ax sec Φ – by cosec Φ = a² – b²

(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 = b²: a²
(b) PF . PG = b²
(c) PF . Pg = a², 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 (x₁, y₁) is T = S₁.

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 (x₁, y₁).

11. Pair of tangents: SS₁ = T²

12. Pole and Polar

(i) Equation of polar of P(x₁, y₁) 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 x² + y² = a² + b².

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.

∠S₁PS₂ = ∠S₁QS₂

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