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# Ellipse - Equation, Formula, Properties, Graphing

## TL;DR
An ellipse is the set of all points in a plane whose distances to two fixed points (called foci) sum to a constant. Its standard equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

## What Is an Ellipse?
An **ellipse** is the locus of all points in a plane whose distances to two fixed points — the **foci** — sum to a constant value. Mathematically, for any point \( P \) on the ellipse:

\[ \|PF_1\| + \|PF_2\| = 2a \]

where \( F_1 \) and \( F_2 \) are the foci and \( 2a \) is the **length of the major axis**.

An ellipse is also a **conic section** — formed by slicing a cone with a plane at a slight angle. A circle is the special case of an ellipse where both foci coincide at the center. The standard equation of an ellipse centered at the origin with major axis along the x-axis is:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b > 0 \]

where \( a \) is the **semi-major axis** length and \( b \) is the **semi-minor axis** length.

## What Are the Parts of an Ellipse?

| Part               | Definition                                                                                              |
|--------------------|---------------------------------------------------------------------------------------------------------|
| **Centre**         | The midpoint of the major axis (and minor axis)                                                        |
| **Foci** (\( F_1, F_2 \)) | Two fixed points whose distance-sum defines the curve                                         |
| **Major axis**     | Longest diameter, length \( 2a \), contains both foci                                                |
| **Minor axis**     | Shortest diameter perpendicular to major axis, length \( 2b \)                                       |
| **Vertex**         | Endpoint of the major axis                                                                               |
| **Co-vertex**      | Endpoint of the minor axis                                                                               |
| **Eccentricity** (\( e \)) | Measure of how "stretched" the ellipse is, \( 0 \le e < 1 \)                               |

## What Is the Equation of an Ellipse?

### Standard Form — Centered at Origin
For an ellipse centered at \( (0,0) \) with major axis along the x-axis:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > b \]
For major axis along the y-axis:
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1, \quad a > b \]

### Shifted Form — Centered at \( (h,k) \)
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

### Relationship Between \( a, b, \) and \( c \)
The distance from the center to each focus is \( c \), where:
\[ c = \sqrt{a^2 - b^2} \]
So the foci sit at \( (\pm c,0) \) for a horizontal ellipse, or \( (0,\pm c) \) for a vertical ellipse.

## How Do You Derive the Ellipse Equation?
The standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) falls out of the focus-pair definition + the distance formula. Here's the derivation.

**Setup.** Place the center at the origin, foci at \( F_1=(-c,0) \) and \( F_2=(c,0) \) and let \( P=(x,y) \) be any point on the ellipse.
By definition, the sum of distances to the foci is \( 2a \):
\[ \sqrt{(x + c)^2 + y^2} + \sqrt{(x - c)^2 + y^2} = 2a \]

**Isolate one radical.**
\[ \sqrt{(x + c)^2 + y^2} = 2a - \sqrt{(x - c)^2 + y^2} \]

**Square both sides.**
\[ (x+c)^2+y^2=4a^2−4a(x−c)^2+y^2+(x−c)^2+y^2 \]

**Solve for remaining radical.**
\[ a(x−c)^2+y^2=a^2−cx \]

**Square again.**
\[ a^2[(x−c)^2+y^2]=a^4−2a^2cx+c^2x^2 \]

**Expand and group.**
\[ (a^2−c^2)x^2+a^2y^2=a^2(a^2−c^2) \]

**Substitute** \( b^2=a^2−c^2 \):
\[ b^2x^2+a^2y^2=a^2b^2 \]

**Divide by \( a^2b^2 \):**
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

## What Are the Properties of an Ellipse?

1. **Two axes of symmetry.** The major and minor axes are both lines of symmetry.
2. **Two foci, constant distance-sum.** For every point \( P \) on the curve, \( \|PF_1\| + \|PF_2\| = 2a \).
3. **Closed, bounded curve.** An ellipse is a finite closed curve, enclosing a finite area.
4. **Area = \( \pi ab \).** Generalizes the circle's \( \pi r^2 \).
5. **Perimeter — no closed-form formula.** Ramanujan's approximation gives \( P \approx \pi \left[ 3(a + b) - \sqrt{(3a + b)(a + 3b)} \right] \), accurate to a few parts per million for moderate eccentricities.
6. **Reflective property.** Any ray emitted from one focus reflects off the ellipse and passes through the other focus.

## How Do You Draw an Ellipse? (The Gardener's Method)
**Materials:** Two pins, a piece of string of length \( 2a \), and a pencil.

**Steps:**
1. Decide the major-axis length \( 2a \) and the focal distance \( 2c \).
2. Press the two pins into the drawing surface, \( 2c \) apart.
3. Tie the string into a loop of total length \( 2a + 2c \).
4. Pull the string taut with the pencil inside the loop and trace a complete circuit.

## What Is Eccentricity?
The **eccentricity** \( e \) of an ellipse measures how elongated it is.  
\[ e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}} \]
-  \( e = 0 \) → perfect circle 
-  \( 0 < e < 1 \) → ellipse  
-  \( e = 1 \) → parabola  
-  \( e > 1 \) → hyperbola  
**Examples:** Earth's orbit has \( e \approx 0.0167 \), Halley's Comet has \( e \approx 0.967 \).

## How Do You Graph an Ellipse?
For an ellipse \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \):
- **Step 1:** Identify the center \( (h,k) \).
- **Step 2:** Determine orientation.
- **Step 3:** Mark the vertices at distance \( a \) from the center.
- **Step 4:** Mark the co-vertices at distance \( b \) from the center.
- **Step 5:** Compute \( c = \sqrt{a^2 - b^2} \) and mark the foci.
- **Step 6:** Connect the four key points with a smooth oval curve.

## Why Are Ellipses Important?  
The ellipse was one of the most important geometric discoveries in the history of science.  
- **Planetary orbits.**  
- **Halley's Comet.**  
- **Whispering galleries.**  
- **Lithotripsy.**  
- **Architecture.**  
- **Engineering — gears.**

## Worked Example of Ellipse
Find the foci of the ellipse \( \frac{x^2}{25} + \frac{y^2}{16} = 1 \).  
**Correct method:**  
\[ c = \sqrt{25 - 16} = 3 \]  
The foci are at \( (\pm 3,0) \).  
**Eccentricity:**  
\[ e = \frac{3}{5} = 0.6 \]

## The Mathematicians Who Shaped Ellipse Theory
- **Apollonius of Perga**  
- **Johannes Kepler  
- **Tycho Brahe**

## Frequently Asked Questions
- **What is an ellipse in simple words?**  
An ellipse is an oval — a stretched circle.  
- **What is the equation of an ellipse?**  
Centered at the origin: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).  
- **What are the foci of an ellipse?**  
Two fixed points inside the ellipse.  
- **What is eccentricity?**  
A number between 0 and 1 that measures how stretched the ellipse is.  
- **How is an ellipse different from a circle?**  
A circle has one center and constant distance to all points.  
- **Where are ellipses used in real life?**  
Planetary orbits, whispering galleries, lithotripsy, architecture, elliptical running tracks.
- **What is the area of an ellipse?**  
\( A = \pi ab \).