# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/12/mathematics/analytic-geometry/ellipse
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/analytic-geometry/ellipse/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Ellipse",
  description: "Discover ellipse properties, foci, and equations. Learn to solve x²/a² + y²/b² = 1 problems with step-by-step examples and interactive visualizations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Analytic Geometry",
};

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

## 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!

<LineEquation
  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: [{ text: "r₁", at: 1, offset: [-1, 0.5, 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: [{ text: "r₂", at: 0, offset: [0.5, 1, 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 }
      ],
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

From the visualization above, notice point <InlineMath math="P" />. The distance from <InlineMath math="P" /> to focus <InlineMath math="F_1" /> (which we call <InlineMath math="r_1" />) plus the distance from <InlineMath math="P" /> to focus <InlineMath math="F_2" /> (which we call <InlineMath math="r_2" />) 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.

<LineEquation
  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,
      smooth: false,
      labels: [{ text: "Major Axis", at: 1, offset: [-1.5, -1, 0] }],
    },
    {
      points: [
        { x: 0, y: -2.8, z: 0 },
        { x: 0, y: 2.8, z: 0 }
      ],
      color: getColor("PINK"),
      showPoints: false,
      smooth: false,
      labels: [{ text: "Minor Axis", at: 1, offset: [1.5, -1, 0] }],
    },
  ]}
  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 <InlineMath math="O" />. All measurements in the ellipse refer to this point.

2. **Foci** (<InlineMath math="F_1" /> and <InlineMath math="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 <InlineMath math="A_1" /> and <InlineMath math="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 <InlineMath math="B_1" /> and <InlineMath math="B_2" />.

5. **Semi-major** (<InlineMath math="a" />) is half the length of the major axis, which is the distance from center to the major axis endpoint.

6. **Semi-minor** (<InlineMath math="b" />) 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 <InlineMath math="a > b" />. If <InlineMath math="a = b" />, 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 <InlineMath math="O(0,0)" />, there are two possible orientations:

<LineEquation
  title="Horizontal Major Axis"
  description={<>Ellipse <InlineMath math="\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1" /> with <InlineMath math="a > b" />.</>}
  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 }
      ],
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

When the major axis is parallel to the <InlineMath math="X" /> axis (horizontal), the ellipse equation is:

<BlockMath math="\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1" />

with the condition <InlineMath math="a > b" />.

<LineEquation
  title="Vertical Major Axis"
  description={<>Ellipse <InlineMath math="\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1" /> with <InlineMath math="a > b" />.</>}
  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 }
      ],
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

When the major axis is parallel to the <InlineMath math="Y" /> axis (vertical), the ellipse equation is:

<BlockMath math="\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1" />

with the condition <InlineMath math="a > b" />.

### Shifted Center

If the ellipse center is not at the origin, but at point <InlineMath math="(h, k)" />, the equation becomes:

<div className="flex flex-col gap-4">
  <BlockMath math="\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \quad \text{(horizontal major axis)}" />
  <BlockMath math="\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1 \quad \text{(vertical major axis)}" />
</div>

Visualization:

<LineEquation
  title="Ellipse with Shifted Center"
  description={<>Ellipse with center at <InlineMath math="(2, -1)" /> 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,
    },
    {
      points: [
        { x: 0, y: -5, z: 0 },
        { x: 0, y: 3, z: 0 }
      ],
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 12]}
  showZAxis={false}
/>

## Important Relationships

There's a formula that always applies to every ellipse:

<BlockMath math="c^2 = a^2 - b^2" />

where <InlineMath math="c" /> is the distance from center to focus.

**Eccentricity** of an ellipse is defined as:

<BlockMath math="e = \frac{c}{a}" />

The eccentricity value of an ellipse is always <InlineMath math="0 < e < 1" />. The closer to <InlineMath math="0" />, the more circular the ellipse becomes. The closer to <InlineMath math="1" />, the more elongated the ellipse becomes.

## Exercises

1. Find the equation of an ellipse with center at <InlineMath math="(0,0)" />, major axis length <InlineMath math="12" /> and minor axis length <InlineMath math="8" />, with horizontal major axis.

2. Given ellipse <InlineMath math="\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 <InlineMath math="(3, -1)" />, foci at <InlineMath math="(3, 2)" /> and <InlineMath math="(3, -4)" />, and minor axis length <InlineMath math="6" />. Find the equation of the ellipse.

4. Find the equation of an ellipse that passes through points <InlineMath math="(4, 3)" /> and <InlineMath math="(6, 2)" /> with center at <InlineMath math="(0, 0)" /> and horizontal major axis.

### Answer Key

1. **Solution**:

   Given:
   - Center at <InlineMath math="(0,0)" />
   - Major axis length = <InlineMath math="12" />, so <InlineMath math="2a = 12" />, thus <InlineMath math="a = 6" />
   - Minor axis length = <InlineMath math="8" />, so <InlineMath math="2b = 8" />, thus <InlineMath math="b = 4" />
   - Horizontal major axis
   
   Ellipse equation with horizontal major axis:
   
   <BlockMath math="\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1" />
   
   Substituting values <InlineMath math="a = 6" /> and <InlineMath math="b = 4" />:
   
   <BlockMath math="\frac{x^2}{36} + \frac{y^2}{16} = 1" />

2. **Solution**:

   From equation <InlineMath math="\frac{x^2}{25} + \frac{y^2}{16} = 1" />:
   - <InlineMath math="a^2 = 25" />, so <InlineMath math="a = 5" />
   - <InlineMath math="b^2 = 16" />, so <InlineMath math="b = 4" />
   
   Since <InlineMath math="a^2 > b^2" />, the major axis is horizontal.
   
   Calculate <InlineMath math="c" />:
   
   <div className="flex flex-col gap-4">
     <BlockMath math="c^2 = a^2 - b^2 = 25 - 16 = 9" />
     <BlockMath math="c = 3" />
   </div>
   
   Foci coordinates: <InlineMath math="(\pm 3, 0)" /> which are <InlineMath math="(-3, 0)" /> and <InlineMath math="(3, 0)" />
   
   Eccentricity:
   
   <BlockMath math="e = \frac{c}{a} = \frac{3}{5} = 0{.}6" />

3. **Solution**:

   Given:
   - Center: <InlineMath math="(3, -1)" />
   - Foci: <InlineMath math="(3, 2)" /> and <InlineMath math="(3, -4)" />
   - Minor axis length = <InlineMath math="6" />, so <InlineMath math="2b = 6" />, thus <InlineMath math="b = 3" />
   
   Since the foci have the same <InlineMath math="x" /> coordinate (<InlineMath math="x = 3" />), the major axis is vertical.
   
   <BlockMath math="\text{Distance between foci} = |2 - (-4)| = 6" />
   
   so <InlineMath math="2c = 6" />, thus <InlineMath math="c = 3" />
   
   Calculate <InlineMath math="a" />:
   
   <div className="flex flex-col gap-4">
     <BlockMath math="c^2 = a^2 - b^2" />
     <BlockMath math="9 = a^2 - 9" />
     <BlockMath math="a^2 = 18" />
     <BlockMath math="a = 3\sqrt{2}" />
   </div>
   
   Ellipse equation with center <InlineMath math="(h,k) = (3,-1)" /> and vertical major axis:
   
   <BlockMath math="\frac{(x-3)^2}{9} + \frac{(y+1)^2}{18} = 1" />

4. **Solution**:

   An ellipse with center <InlineMath math="(0,0)" /> and horizontal major axis has the equation:
   
   <BlockMath math="\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1" />
   
   Substituting point <InlineMath math="(4,3)" />:
   
   <BlockMath math="\frac{16}{a^2} + \frac{9}{b^2} = 1 \quad \text{...(1)}" />
   
   Substituting point <InlineMath math="(6,2)" />:
   
   <BlockMath math="\frac{36}{a^2} + \frac{4}{b^2} = 1 \quad \text{...(2)}" />
   
   Let <InlineMath math="u = \frac{1}{a^2}" /> and <InlineMath math="v = \frac{1}{b^2}" />, then:
   
   <div className="flex flex-col gap-4">
     <BlockMath math="16u + 9v = 1 \quad \text{...(1)}" />
     <BlockMath math="36u + 4v = 1 \quad \text{...(2)}" />
   </div>
   
   From equation (1): <InlineMath math="v = \frac{1-16u}{9}" />
   
   Substituting into equation (2):
   
   <div className="flex flex-col gap-4">
     <BlockMath math="36u + 4 \cdot \frac{1-16u}{9} = 1" />
     <BlockMath math="36u + \frac{4-64u}{9} = 1" />
     <BlockMath math="324u + 4 - 64u = 9" />
     <BlockMath math="260u = 5" />
     <BlockMath math="u = \frac{1}{52}" />
   </div>
   
   So <InlineMath math="a^2 = 52" />
   
   Substituting back:
   
   <div className="flex flex-col gap-4">
     <BlockMath 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}" />
   </div>
   
   So <InlineMath math="b^2 = 13" />
   
   Ellipse equation: <InlineMath math="\frac{x^2}{52} + \frac{y^2}{13} = 1" />