# Nakafa Learning Content
> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.
URL: https://nakafa.com/en/subjects/mathematics/analytic-geometry/ellipse
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/analytic-geometry/ellipse/en.mdx
Learn ellipse properties, foci, and equations, then solve x²/a² + y²/b² = 1 problems with worked examples.
---
## What is an Ellipse?
Ever seen the shape of planets orbiting the sun? Or the shadow of a circle when viewed from the side? Well, shapes like those are called **ellipses**! An ellipse isn't just a "flattened" circle, but there's a cool mathematical definition behind it.
So here's the thing, an ellipse is a collection of points where **the sum of distances to two specific points is always the same**. These two specific points are called **foci**. Just imagine you have two nails and a string. If you tie the string to both nails, then pull a pencil until the string is tight and draw a complete curve, the curve formed is an ellipse!
Component: LineEquation
Props:
- title: Basic Ellipse Concept
- description: Ellipse with two foci and several points showing constant sum of distances.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
const a = 4;
const b = 2.5;
return {
x: a * Math.cos(angle),
y: b * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 }
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O", at: 0, offset: [-0.3, -0.3, 0] }],
},
{
points: (() => {
const a = 4, b = 2.5;
const c = Math.sqrt(a*a - b*b);
return [
{ x: -c, y: 0, z: 0 },
{ x: c, y: 0, z: 0 }
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "F₁", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "F₂", at: 1, offset: [0.5, -0.5, 0] }
],
},
{
points: (() => {
const angle = Math.PI / 3;
const a = 4, b = 2.5;
return [{
x: a * Math.cos(angle),
y: b * Math.sin(angle),
z: 0,
}];
})(),
color: getColor("LIME"),
showPoints: true,
labels: [{ text: "P", at: 0, offset: [0.5, 0.3, 0] }],
},
{
points: (() => {
const angle = Math.PI / 3;
const a = 4, b = 2.5;
const c = Math.sqrt(a*a - b*b);
const px = a * Math.cos(angle);
const py = b * Math.sin(angle);
return [
{ x: -c, y: 0, z: 0 },
{ x: px, y: py, z: 0 }
];
})(),
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels ... [truncated; 1839 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
From the visualization above, notice point $$P$$. The distance from $$P$$ to focus $$F_1$$ (which we call $$r_1$$) plus the distance from $$P$$ to focus $$F_2$$ (which we call $$r_2$$) will always be the same for all points on the ellipse. This is the **fundamental characteristic of an ellipse**!
Visible text: From the visualization above, notice point . The distance from to focus (which we call ) plus the distance from to focus (which we call ) will always be the same for all points on the ellipse. This is the **fundamental characteristic of an ellipse**!
## Ellipse Components
Before we discuss the formulas, let's get familiar with the important parts of an ellipse. Each part has its own role in determining the shape of the ellipse.
Component: LineEquation
Props:
- title: Ellipse Components
- description: Important parts of an ellipse with horizontal major axis.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
const a = 4;
const b = 2.8;
return {
x: a * Math.cos(angle),
y: b * Math.sin(angle),
z: 0,
};
}),
color: getColor("INDIGO"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 }
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "Center", at: 0, offset: [-0.8, -0.5, 0] }],
},
{
points: (() => {
const a = 4, b = 2.8;
const c = Math.sqrt(a*a - b*b);
return [
{ x: -c, y: 0, z: 0 },
{ x: c, y: 0, z: 0 }
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "F₁", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "F₂", at: 1, offset: [0.5, -0.5, 0] }
],
},
{
points: [
{ x: -4, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 }
],
color: getColor("TEAL"),
showPoints: true,
labels: [
{ text: "A₁", at: 0, offset: [-0.5, 0.5, 0] },
{ text: "A₂", at: 1, offset: [0.5, 0.5, 0] }
],
},
{
points: [
{ x: 0, y: -2.8, z: 0 },
{ x: 0, y: 2.8, z: 0 }
],
color: getColor("EMERALD"),
showPoints: true,
labels: [
{ text: "B₁", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "B₂", at: 1, offset: [-0.5, 0.5, 0] }
],
},
{
points: [
{ x: -4, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 }
],
color: getColor("YELLOW"),
showPoints: false ... [truncated; 1473 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
Here are the components you need to know:
1. **Ellipse center** is the midpoint of the ellipse, usually written with letter $$O$$. All measurements in the ellipse refer to this point.
2. **Foci** ($$F_1$$ and $$F_2$$) are two fixed points that serve as reference for the ellipse definition. The distance between the two foci is called the focal distance.
3. **Major axis** is the longest line that passes through the ellipse center and both foci. Its endpoints are points $$A_1$$ and $$A_2$$.
4. **Minor axis** is the shortest line that passes through the ellipse center and is perpendicular to the major axis. Its endpoints are points $$B_1$$ and $$B_2$$.
5. **Semi-major** ($$a$$) is half the length of the major axis, which is the distance from center to the major axis endpoint.
6. **Semi-minor** ($$b$$) is half the length of the minor axis, which is the distance from center to the minor axis endpoint.
Visible text: 1. **Ellipse center** is the midpoint of the ellipse, usually written with letter . All measurements in the ellipse refer to this point.
2. **Foci** ( and ) are two fixed points that serve as reference for the ellipse definition. The distance between the two foci is called the focal distance.
3. **Major axis** is the longest line that passes through the ellipse center and both foci. Its endpoints are points and .
4. **Minor axis** is the shortest line that passes through the ellipse center and is perpendicular to the major axis. Its endpoints are points and .
5. **Semi-major** () is half the length of the major axis, which is the distance from center to the major axis endpoint.
6. **Semi-minor** () is half the length of the minor axis, which is the distance from center to the minor axis endpoint.
> Remember, in an ellipse we always have $$a > b$$. If $$a = b$$, the shape becomes a circle!
Visible text: > Remember, in an ellipse we always have . If , the shape becomes a circle!
## Ellipse Equations
Now, let's get into the fun part: how to write an ellipse in mathematical equation form. There are several forms depending on position and orientation.
### Center at Origin
If the ellipse center is at $$O(0,0)$$, there are two possible orientations:
Visible text: If the ellipse center is at , there are two possible orientations:
Component: LineEquation
Props:
- title: Horizontal Major Axis
- description: Ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ with $$a > b$$.
Visible text: Ellipse with .
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
const a = 4.5;
const b = 3;
return {
x: a * Math.cos(angle),
y: b * Math.sin(angle),
z: 0,
};
}),
color: getColor("VIOLET"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 }
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O(0,0)", at: 0, offset: [-0.8, -0.5, 0] }],
},
{
points: (() => {
const a = 4.5, b = 3;
const c = Math.sqrt(a*a - b*b);
return [
{ x: -c, y: 0, z: 0 },
{ x: c, y: 0, z: 0 }
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "F₁", at: 0, offset: [-0.5, 0.5, 0] },
{ text: "F₂", at: 1, offset: [0.5, 0.5, 0] }
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 4.5, y: 0, z: 0 }
],
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "a", at: 1, offset: [0, -0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 3, z: 0 }
],
color: getColor("EMERALD"),
showPoints: false,
smooth: false,
labels: [{ text: "b", at: 1, offset: [0.5, 0, 0] }],
},
{
points: [
{ x: -6, y: 0, z: 0 },
{ x: 6, y: 0, z: 0 }
],
color: getColor("ROSE"),
showPoints: false,
smooth: false,
},
{
points: [
{ x: 0, y: -4, z: 0 },
{ x: 0, y: 4, z: 0 }
],
c ... [truncated; 1262 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
When the major axis is parallel to the $$X$$ axis (horizontal), the ellipse equation is:
Visible text: When the major axis is parallel to the axis (horizontal), the ellipse equation is:
```math
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
```
with the condition $$a > b$$.
Visible text: with the condition .
Component: LineEquation
Props:
- title: Vertical Major Axis
- description: Ellipse $$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$ with $$a > b$$.
Visible text: Ellipse with .
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
const a = 4.5;
const b = 3;
return {
x: b * Math.cos(angle),
y: a * Math.sin(angle),
z: 0,
};
}),
color: getColor("TEAL"),
showPoints: false,
},
{
points: [
{ x: 0, y: 0, z: 0 }
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "O(0,0)", at: 0, offset: [-0.8, -0.5, 0] }],
},
{
points: (() => {
const a = 4.5, b = 3;
const c = Math.sqrt(a*a - b*b);
return [
{ x: 0, y: -c, z: 0 },
{ x: 0, y: c, z: 0 }
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "F₁", at: 0, offset: [0.5, -0.5, 0] },
{ text: "F₂", at: 1, offset: [0.5, 0.5, 0] }
],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 }
],
color: getColor("EMERALD"),
showPoints: false,
smooth: false,
labels: [{ text: "b", at: 1, offset: [0, -0.5, 0] }],
},
{
points: [
{ x: 0, y: 0, z: 0 },
{ x: 0, y: 4.5, z: 0 }
],
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "a", at: 1, offset: [0.5, 0, 0] }],
},
{
points: [
{ x: -5, y: 0, z: 0 },
{ x: 5, y: 0, z: 0 }
],
color: getColor("ROSE"),
showPoints: false,
smooth: false,
},
{
points: [
{ x: 0, y: -6, z: 0 },
{ x: 0, y: 6, z: 0 }
],
col ... [truncated; 1260 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
When the major axis is parallel to the $$Y$$ axis (vertical), the ellipse equation is:
Visible text: When the major axis is parallel to the axis (vertical), the ellipse equation is:
```math
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
```
with the condition $$a > b$$.
Visible text: with the condition .
### Shifted Center
If the ellipse center is not at the origin, but at point $$(h, k)$$, the equation becomes:
Visible text: If the ellipse center is not at the origin, but at point , the equation becomes:
Component: MathContainer
Children:
```math
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \text{(horizontal major axis)}
```
```math
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \quad \text{(vertical major axis)}
```
Visualization:
Component: LineEquation
Props:
- title: Ellipse with Shifted Center
- description: Ellipse with center at $$(2, -1)$$ and horizontal major axis.
Visible text: Ellipse with center at and horizontal major axis.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
const a = 3.5;
const b = 2.5;
const h = 2, k = -1;
return {
x: h + a * Math.cos(angle),
y: k + b * Math.sin(angle),
z: 0,
};
}),
color: getColor("SKY"),
showPoints: false,
},
{
points: [
{ x: 2, y: -1, z: 0 }
],
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "(2,-1)", at: 0, offset: [-0.8, 0.5, 0] }],
},
{
points: (() => {
const a = 3.5, b = 2.5;
const c = Math.sqrt(a*a - b*b);
const h = 2, k = -1;
return [
{ x: h - c, y: k, z: 0 },
{ x: h + c, y: k, z: 0 }
];
})(),
color: getColor("CYAN"),
showPoints: true,
labels: [
{ text: "F₁", at: 0, offset: [-0.5, 0.5, 0] },
{ text: "F₂", at: 1, offset: [0.5, 0.5, 0] }
],
},
{
points: [
{ x: 2, y: -1, z: 0 },
{ x: 5.5, y: -1, z: 0 }
],
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "a", at: 1, offset: [0, -0.5, 0] }],
},
{
points: [
{ x: 2, y: -1, z: 0 },
{ x: 2, y: 1.5, z: 0 }
],
color: getColor("EMERALD"),
showPoints: false,
smooth: false,
labels: [{ text: "b", at: 1, offset: [0.5, 0, 0] }],
},
{
points: [
{ x: -3, y: 0, z: 0 },
{ x: 7, y: 0, z: 0 }
],
color: getColor("ROSE"),
showPoints: false,
smooth: false, ... [truncated; 1325 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false
## Important Relationships
There's a formula that always applies to every ellipse:
```math
c^2 = a^2 - b^2
```
where $$c$$ is the distance from center to focus.
Visible text: where is the distance from center to focus.
**Eccentricity** of an ellipse is defined as:
```math
e = \frac{c}{a}
```
The eccentricity value of an ellipse is always $$0 < e < 1$$. The closer to $$0$$, the more circular the ellipse becomes. The closer to $$1$$, the more elongated the ellipse becomes.
Visible text: The eccentricity value of an ellipse is always . The closer to , the more circular the ellipse becomes. The closer to , the more elongated the ellipse becomes.
## Exercises
1. Find the equation of an ellipse with center at $$(0,0)$$, major axis length $$12$$ and minor axis length $$8$$, with horizontal major axis.
2. Given ellipse $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$. Find the coordinates of the foci and the eccentricity of the ellipse.
3. An ellipse has center at $$(3, -1)$$, foci at $$(3, 2)$$ and $$(3, -4)$$, and minor axis length $$6$$. Find the equation of the ellipse.
4. Find the equation of an ellipse that passes through points $$(4, 3)$$ and $$(6, 2)$$ with center at $$(0, 0)$$ and horizontal major axis.
Visible text: 1. Find the equation of an ellipse with center at , major axis length and minor axis length , with horizontal major axis.
2. Given ellipse . Find the coordinates of the foci and the eccentricity of the ellipse.
3. An ellipse has center at , foci at and , and minor axis length . Find the equation of the ellipse.
4. Find the equation of an ellipse that passes through points and with center at and horizontal major axis.
### Answer Key
1. **Solution**:
Given:
- Center at $$(0,0)$$
- Major axis length is $$12$$, so $$2a = 12$$, thus $$a = 6$$
- Minor axis length is $$8$$, so $$2b = 8$$, thus $$b = 4$$
- Horizontal major axis
Ellipse equation with horizontal major axis:
```math
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
```
Substituting values $$a = 6$$ and $$b = 4$$:
```math
\frac{x^2}{36} + \frac{y^2}{16} = 1
```
2. **Solution**:
From equation $$\frac{x^2}{25} + \frac{y^2}{16} = 1$$:
- $$a^2 = 25$$, so $$a = 5$$
- $$b^2 = 16$$, so $$b = 4$$
Since $$a^2 > b^2$$, the major axis is horizontal.
Calculate $$c$$:
```math
c^2 = a^2 - b^2 = 25 - 16 = 9
```
```math
c = 3
```
Foci coordinates: $$(\pm 3, 0)$$ which are $$(-3, 0)$$ and $$(3, 0)$$
Eccentricity:
```math
e = \frac{c}{a} = \frac{3}{5} = 0{.}6
```
3. **Solution**:
Given:
- Center: $$(3, -1)$$
- Foci: $$(3, 2)$$ and $$(3, -4)$$
- Minor axis length is $$6$$, so $$2b = 6$$, thus $$b = 3$$
Since the foci have the same $$x$$ coordinate ($$x = 3$$), the major axis is vertical.
```math
\text{Distance between foci} = |2 - (-4)| = 6
```
so $$2c = 6$$, thus $$c = 3$$
Calculate $$a$$:
```math
c^2 = a^2 - b^2
```
```math
9 = a^2 - 9
```
```math
a^2 = 18
```
```math
a = 3\sqrt{2}
```
Ellipse equation with center $$(h,k) = (3,-1)$$ and vertical major axis:
```math
\frac{(x-3)^2}{9} + \frac{(y+1)^2}{18} = 1
```
4. **Solution**:
An ellipse with center $$(0,0)$$ and horizontal major axis has the equation:
```math
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
```
Substituting point $$(4,3)$$:
```math
\frac{16}{a^2} + \frac{9}{b^2} = 1 \quad \text{...(1)}
```
Substituting point $$(6,2)$$:
```math
\frac{36}{a^2} + \frac{4}{b^2} = 1 \quad \text{...(2)}
```
Let $$u = \frac{1}{a^2}$$ and $$v = \frac{1}{b^2}$$, then:
```math
16u + 9v = 1 \quad \text{...(1)}
```
```math
36u + 4v = 1 \quad \text{...(2)}
```
From equation ($$1$$): $$v = \frac{1-16u}{9}$$
Substituting into equation ($$2$$):
```math
36u + 4 \cdot \frac{1-16u}{9} = 1
```
```math
36u + \frac{4-64u}{9} = 1
```
```math
324u + 4 - 64u = 9
```
```math
260u = 5
```
```math
u = \frac{1}{52}
```
So $$a^2 = 52$$
Substituting back:
```math
v = \frac{1-16 \cdot \frac{1}{52}}{9} = \frac{1-\frac{16}{52}}{9} = \frac{\frac{36}{52}}{9} = \frac{4}{52} = \frac{1}{13}
```
So $$b^2 = 13$$
Ellipse equation: $$\frac{x^2}{52} + \frac{y^2}{13} = 1$$
Visible text: 1. **Solution**:
Given:
- Center at
- Major axis length is , so , thus
- Minor axis length is , so , thus
- Horizontal major axis
Ellipse equation with horizontal major axis:
Substituting values and :
2. **Solution**:
From equation :
- , so
- , so
Since , the major axis is horizontal.
Calculate :
Foci coordinates: which are and
Eccentricity:
3. **Solution**:
Given:
- Center:
- Foci: and
- Minor axis length is , so , thus
Since the foci have the same coordinate (), the major axis is vertical.
so , thus
Calculate :
Ellipse equation with center and vertical major axis:
4. **Solution**:
An ellipse with center and horizontal major axis has the equation:
Substituting point :
Substituting point :
Let and , then:
From equation ():
Substituting into equation ():
So
Substituting back:
So
Ellipse equation: