Ellipse Formula Sheet

An ellipse is an oval-shaped curve, defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. It can be thought of as a stretched circle along its major and minor axes.

Neetesh Kumar | June 25, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Standard Equation & Definition
• 2. Another Form of Ellipse
• 3. General Equation of an Ellipse
• 4. Position of a Point W.R.T. an Ellipse
• 5. Auxiliary Circle
• 6. Parametric Representation
• 7. Line and an Ellipse
• 8. Tangent to the Ellipse
• 9. Normal to the Ellipse
• 10. Chord of Contact
• 11. Pair of Tangents
• 12. Director Circle
• 13. Equation of Chord with Midpoint
• 14. Important Highlights

1. Standard Equation & Definition

The standard equation of an ellipse referred to its principal axis along the co-ordinate axis is

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where a > b and $b^2 = a^2(1-e^2)$ $\Rightarrow$ $a^2 – b^2 = a^2 e^2$.

where e is the eccentricity (0 < e < 1).

Foci

S = (ae, 0) and S' = (–ae, 0)

Equation of Directrices

$x = \frac{a}{e}$ and $x = -\frac{a}{e}$

Vertices

A = (a, 0) and A' = (–a, 0)

Major Axis

The line segment A'A in which the foci S' and S lie is of length 2a and is called the ellipse's major axis (a > b). The point of intersection of the major axis with the directrix is called the foot of the directrix (Z) $(\plusmn \frac{a}{e}, 0)$

Minor Axis

The y-axis intersects the ellipse in the points ( B' = (0, –b) ) and ( B = (0, b) ). The line segment ( B'B ) of length ( 2b ) is called the ellipse's minor axis.

Centre

The point which bisects every chord of the conic drawn through it is called the centre of the conic. ( C = (0,0) ) is the centre of the ellipse.

Diameter

A chord of the conic which passes through the centre is called a diameter of the conic.

Focal Chord

A chord which passes through a focus is called a focal chord.

Double Ordinate

A chord perpendicular to the major axis is called a double ordinate with respect to major axis as diameter.

Latus Rectum

The focal chord perpendicular to the major axis is called the latus rectum.

• Length of latus rectum $LL' = \frac{2b^2}{a} = 2a(1 - e^2)$
• Equation of latus rectum ( x = ± ae )
• Ends of the latus rectum are $L( ae, \frac{b^2}{a}),$ $L'(ae, -\frac{b^2}{a}),$ $L_1(-ae, \frac{b^2}{a})$ and $L_1'(-ae, -\frac{b^2}{a})$

2. Another Form of Ellipse

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \space where \space a < b$

Major Axis

AA' = Minor axis = 2a

Minor Axis

BB' = Major axis = 2b

Latus Rectum

$LL' = \frac{2a^2}{b}$ Equation ( y = ± $\frac{b}{e}$ )

Eccentricity

$e = \sqrt{1 - \frac{a^2}{b^2}}$

3. General Equation of an Ellipse

Let (a,b) be the focus (S), and (lx + my + n = 0) is the equation of directrix. Let P (x,y) be any point on the ellipse. Then by definition $SP = e PM$ $(x – a)^2 + (y – b)^2 = e^2 \frac{(lx + my + n)^2}{l^2 + m^2}$

4. Position of a Point W.R.T. an Ellipse

The point P $(x_1, y_1) lies outside, inside, or on the ellipse according as $\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1 > < or = 0.$

5. Auxiliary Circle

A circle described on the major axis as diameter is called the auxiliary circle.

Eccentric Angle

If (Q) is a point on the auxiliary circle $x^2 + y^2 = a^2$ such that (QP) produced is perpendicular to the x-axis, then (P) and (Q) are called corresponding points on the ellipse and the auxiliary circle, respectively. ( theta ) is called the eccentric angle of the point (P) on the ellipse.

6. Parametric Representation

The equations $x = a cos \theta$ and $y = b sin\theta$ together represent the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where ( theta ) is a parameter (eccentric angle).

7. Line and an Ellipse

The line ( y = mx + c ) meets the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ in two real points, coincident or imaginary, according to ( c^2 lesser a^2m^2 + b^2 ).

Tangent to the Ellipse

If $c^2 = a^2m^2 + b^2$, the line ( y = mx + c ) is tangent to the ellipse.

8. Tangent to the Ellipse

Point Form

The equation of tangent to the given ellipse at its point $(x_1, y_1)$ is $\frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = 1$

Slope Form

The equation of a tangent to the given ellipse whose slope is 'm' is $y = mx ± \sqrt{a^2 m^2 + b^2}$

Parametric Form

The equation of a tangent to the given ellipse at its point $(a cos \theta, b sin \theta)$, is $\frac{x cos \theta}{a} + \frac{y sin \theta}{b} = 1$

9. Normal to the Ellipse

Point Form

The equation of the normal to the given ellipse at ((x_1, y_1)) is $\frac{a^2 x}{x_1} - \frac{b^2 y}{y_1} = a^2 - b^2$

Slope Form

Equation of a normal to the given ellipse whose slope is 'm' is $y = mx - \frac{a^2 - b^2}{a m^2 + b^2}$

Parametric Form

The equation of the normal to the given ellipse at the point ((a cos theta, b sin theta)) is $a sec \theta - b csc \theta = a^2 - b^2$

10. Chord of Contact

If ( PA ) and ( PB ) are the tangents from point P $(x_1,y_1)$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ then the equation of the chord of contact ( AB ) is $\frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} = 1$

11. Pair of Tangents

If P $(x_1,y_1)$ lies outside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ then a pair of tangents ( PA ) and ( PB ) can be drawn to it from ( P ). The equation of a pair of tangents is $(SS_1 = T^2)$, where
$S = \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1$
$T = \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} - 1$

12. Director Circle

The Locus of the point of intersection of the tangents that meet at right angles is called the Director Circle. The equation to this locus is $x^2 + y^2 = a^2 + b^2$

13. Equation of Chord with Midpoint

The equation of the chord of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$ whose midpoint be $(x_1,y_1)$ is (T = S$_1$).
where
$T = \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} - 1$ $S_1 = \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1$

14. Important Highlights

Tangent and Normal

At a point ( P ) on the ellipse, the tangent and normal bisect the external and internal angles between the focal distances of ( P ).

Reflection Property

Rays from one focus are reflected through the other focus.

Conormal Points

If ( A(a), B(b), C(g), D(d) ) are conormal points, then sum of their eccentric angles is an odd multiple of ( pi ).

Concyclic Points

If ( A(a), B(b), C(g), D(d) ) are four concyclic points, then sum of their eccentric angles is an even multiple of ( pi ).

Perpendicular Segments

The product of the lengths of the perpendicular segments from the foci on any tangent to the ellipse is ( b^2 ) and the feet of these perpendiculars lie on its auxiliary circle.

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