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Equations and Properties of Ellipses

CSCA Equations and Properties of Ellipses study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Equations and Properties of Ellipses

Ellipses are a focal point of the Analytic Geometry section in the CSCA exam. The key to mastering ellipses lies in **"Positioning"** (determining which axis the foci lie on) and **"Quantifying"** (understanding the relationship between $a, b, c$).

1. Definition

An ellipse is the locus of points in a plane such that the sum of the distances to two fixed points $F_1, F_2$ (**Foci**) is a constant $2a$.

* Condition: $2a > |F_1F_2|$ (The constant must be greater than the focal distance).

* Focal Length: $|F_1F_2| = 2c$.

* Mathematical Expression: $|PF_1| + |PF_2| = 2a$.

2. Standard Equations

The equation of an ellipse depends on the coordinate axis where the foci are located. Regardless of the location, the following relationship always holds:

$$a^2 = b^2 + c^2 \quad (a > b > 0, a > c > 0)$$

**Note**: In an ellipse, $a$ (semi-major axis) is the longest segment, which differs from a hyperbola.

| Type | **Foci on x-axis (Horizontal)** | **Foci on y-axis (Vertical)** |

| :--- | :---: | :---: |

| **Graph** | | (Similar, but major axis is vertical) |

| **Standard Equation** | $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ | $$\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$$ |

| **Foci Coordinates** | $F(\pm c, 0)$ | $F(0, \pm c)$ |

| **Vertices** | $(\pm a, 0), (0, \pm b)$ | $(0, \pm a), (\pm b, 0)$ |

| **How to Identify** | **Denominator of $x^2$ is larger** | **Denominator of $y^2$ is larger** |

3. Geometric Properties

Taking the horizontal ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ as an example:

1. **Range**: $-a \le x \le a$, $-b \le y \le b$. The ellipse is bounded within a rectangle.

2. **Symmetry**: Symmetric about the x-axis, y-axis, and the origin.

3. **Eccentricity ($e$)**:

$$e = \frac{c}{a} \quad (0 < e < 1)$$

* $e$ describes the "flatness" of the ellipse.

* $e \to 0$: The ellipse approaches a **circle**.

* $e \to 1$: The ellipse becomes **flatter**.

4. **Major and Minor Axes**:

* Length of Major Axis = $2a$, Length of Minor Axis = $2b$.

4. Examples

**Example 1**: Given the ellipse equation $25x^2 + 9y^2 = 225$, find the coordinates of the foci and the eccentricity.

**Solution**:

1. **Standard Form**: Divide both sides by 225 to get $\frac{x^2}{9} + \frac{y^2}{25} = 1$.

2. **Identify Focus**: Since $25 > 9$ and 25 is under $y^2$, the foci are on the **y-axis**.

3. **Determine Parameters**: $a^2=25 \Rightarrow a=5$, $b^2=9 \Rightarrow b=3$.

4. **Calculate c**: $c = \sqrt{a^2 - b^2} = \sqrt{25-9} = 4$.

5. **Conclusion**: Foci are $(0, \pm 4)$, Eccentricity $e = \frac{c}{a} = \frac{4}{5} = 0.8$.

**Example 2**: The eccentricity of an ellipse is $\frac{1}{2}$ and the length of the major axis is 4. Find the standard equation.

**Solution**:

From the problem, $2a = 4 \Rightarrow a = 2$.

Since $e = \frac{c}{a} = \frac{1}{2}$, we get $c = 1$.

Then $b^2 = a^2 - c^2 = 4 - 1 = 3$.

Since the focus location is not specified, consider two cases:

* Foci on x-axis: $\frac{x^2}{4} + \frac{y^2}{3} = 1$

* Foci on y-axis: $\frac{y^2}{4} + \frac{x^2}{3} = 1$

5. Study Tips

* **Analyze the Equation First**: Always check which denominator is larger to determine the axis of the foci.

* **Memorize the Relationship**: For ellipses, $a^2 = b^2 + c^2$ ($a$ is the hypotenuse/longest).

* **Distinguish from Hyperbola**: For hyperbolas, $c^2 = a^2 + b^2$ ($c$ is the hypotenuse). Do not confuse them.