Ellipse

An Ellipse is a curved shape that represents the path traced by a point moving in a plane in such a way that the combined distance from two fixed points remains constant in the plane, known as foci, remains constant.

The two fixed points of an ellipse are known as its foci. Here, the foci of an Ellipse are F₁ and F₂.

1.0Ellipse Definition

An Ellipse is a geometric figure defined as the set of all points in a plane where the total distance to two fixed points, known as the foci, remain constant. In simpler terms, an ellipse is a closed curve that resembles a flattened circle. It has two axes: a major axis, which is the longest diameter of the ellipse, and a minor axis, which is the shortest cross-sectional diameter, that intersects the major axis at a right angle. The equation of an ellipse in standard form is:

• Point (h, k) denotes the coordinates of the ellipse’s center..
• a is the length of the semi-major axis (half of the major axis length).
• b represents the measure of the semi-minor axis (half of the minor axis length).

or

Ellipse is the locus of a point, ratio of whose distance from a fixed point and a fixed line is constant and less than 1.

2.0Ellipse General Equation

The general equation of an ellipse with its center at the origin, semi-major axis a along the x-axis and semi-minor axis b along the y-axis is given by:

This equation represents all points on the ellipse where the sum of the squares of their distances from the center is equal to 1. The measure of the semi-major axis a and semi-minor axis b determine the size and shape of the ellipse.

Another form of Equation of Ellipse

Ellipse Equation can be represented as ax² + 2hxy + by² + 2gx + 2fy + c = 0

if Δ ≠ 0, h² < ab

where: Δ = abc + 2fgh – af² – bg² + ch²

3.0Standard Equations of Ellipse

The standard equation of an ellipse is given by:

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, Where 0 < b < a.

This is also called the standard form of the equation of the horizontal ellipse.

The standard form of the equation of a vertical ellipse is given by $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$_,  where 0 < a < b.

4.0Properties of Ellipse

1. Axes: The major and minor axes of an ellipse are perpendicular to each other. The major axis is the ellipse's longest diameter, while the minor axis is its shortest diameter.
2. Eccentricity: The ellipse eccentricity is denoted by e. It is a measure of how elongated or stretched the ellipse is. It is defined as the ratio of the distance between the foci to the major axis’s length. The ellipse eccentricity is always less than 1.
3. Center: The center of an ellipse is the point at which the major and minor axes intersect.
4. Vertex: The vertices of an ellipse are the points where the ellipse intersects its major axis. 
5. Focus: The foci of an ellipse are two fixed points inside the ellipse such that the total distance from any point on the ellipse to both foci is constant. 
6. Latus Rectum: The Ellipse Latus Rectum is a line segment that extends from one focus of the ellipse and is perpendicular to the major axis. The  Ellipse latus rectum can also be expressed using eccentricity e and the semi-major axis a as l = 2b²/a​. This formula is fundamental in describing the geometry and characteristics of the ellipse.

5.0Equation of Tangent to the Ellipse 

A tangent to an ellipse is a straight line that touches the curve at exactly one point.

Slope Form

Equation of Tangent to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, for all m ∈ R at point  is

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

$y=mx\pm \sqrt{a^2{m}^2+b^2}$  

 Cartesian Form

The equation of Tangent to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at point (x₁, y₁) is:

$\frac{x\cdot x_1}{a^2}+\frac{y\cdot y_1}{b^2}=1$

Parametric Form

The equation of Tangent to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, at the point (acosϴ, bsinϴ) is:

$\frac{x\cos \theta }{a}+\frac{y\sin \theta }{b}=1$

Note: For y = mx + c to be tangent to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is c² = a²m² + b². This is a condition of tangency. 

6.0Equation of Normal to the Ellipse 

The normal to an Ellipse at a given point is a line perpendicular to the tangent at that point. It provides insight into the curvature of the ellipse at that specific location.

Slope Form

Equation of Normal to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ for all m ∈ R at is:

$y=mx-\frac{m\left(a^2-b^2\right)}{\sqrt{a^2+b^2{m}^2}}$

Cartesian Form

The equation of Normal to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at point (x₁, y₁) is:

$\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2$

Parametric Form

The equation of Normal to Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at point (acosϴ, bsinϴ) is:

$\frac{ax}{\cos \theta }-\frac{by}{\sin \theta }=a^2-b^2$

7.0Solved Examples

Question 1: Find the equation of an ellipse whose focus is (2, 3) eccentricity is $\frac{1}{3}$ and the directrix x +2y –1 = 0

Solution:  Let P (x, y) be a point on the ellipse whose focus is s(2,3), and the directrix x+2y–1 = 0

Then, by definition, SP = e PM

⇒ (SP)²  = e² (PM)² 

⇒ (x–2)² + (y –3)² = $\frac{1}{9}{\left\{\frac{x+2y-1}{\sqrt{5}}\right\}}^2$

⇒ 45 (x² + y²– 4x –6y + 13) = x² + 4y² + 1 + 4xy –2x–4y

⇒ 44x² + 41 y²– 4xy – 178 x – 266y + 584 = 0

Which is the required equation of the ellipse. 

Question 2: Find all parameters of the ellipse 4x² + 9y² = 36

Solution:   $\frac{x^2}{9}+\frac{y^2}{4}=1$

This is of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

So a = 3,b = 2, e =$\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{4}{9}}=\frac{\sqrt{5}}{3}$

$\Rightarrow \mathrm{ae}=\sqrt{5},\frac{\mathrm{a}}{\mathrm{e}}=\frac{9}{\sqrt{5}}$

center (0,0), Vertex (± a,0) = (±3, 0)

Foci : (± ae, 0) = $(\pm \sqrt{5},0)$

Ends of latus rectum

$\left(\pm \mathrm{ae},\pm \frac{{\mathrm{b}}^2}{\mathrm{a}}\right)=\left(\pm \sqrt{5},\pm \frac{4}{3}\right)$

Directrix  x =

$x=\pm \frac{a}{e}=x=\pm \frac{9}{\sqrt{5}}$

Question 3 : Find equation of tangent to an ellipse 3x² + 4y² = 12, Parallel to the line y + 2x = 4

Solution: Equation of ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$

So a² = 4, b² = 3

Line y + 2x = 4 y= – 2x + 4

⇒ m = –2

Equation of tangent to an ellipse

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

$y=$ $-2x\pm \sqrt{4(-2{)}^2+3}$

$y=-2x\pm \sqrt{19}$

Question 4: Find the equation of the tangents to the ellipse x² +16y² = 16, each one of which makes an angle of 60° with the x-axis

Solution : We have x² + 16y² = 16 ⇒ $\frac{x^2}{4^2}+\frac{y^2}{1^2}=1$

Where a² = 16, b² = 1

Slope of tangent m = tan 60° = $\sqrt{3}$

So the equation of tangent is.

$y=mx\pm \sqrt{a^2{m}^2+b^2}\Rightarrow y=\sqrt{3}x\pm \sqrt{16\times 3+1}$

$\Rightarrow y=\sqrt{3}x\pm \sqrt{48+1}\Rightarrow y=\sqrt{3}x\pm 7$

Question 5: Find the equation of the normal of the ellipse 9x² + 16y² = 288 at the point (4,3) 

Solution: Ellipse is $\frac{x^2}{32}+\frac{y^2}{18}=1$  , x₁ = 4, y₁ = 3

Equation of normal is = $\frac{a^{2}x}{x_1}-\frac{b^{2}y}{y_1}=a^2-b^2$

=$\frac{32x}{4}-\frac{18y}{3}$

= 32 –18 ⇒ 8x –6y = 14

⇒ 4x –3y = 7