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Equations and Properties of Hyperbolas

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

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Equations and Properties of Hyperbolas

The hyperbola is the most unique of the three conic sections, especially due to its **asymptotes**. For the CSCA exam, the keys are determining the **opening direction** (which axis the foci lie on) based on the equation and mastering the **Pythagorean relationship** between $a, b, and c$.

1. Definition

A hyperbola is the locus of points in a plane such that the **absolute value** of the difference 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 less than the focal distance).

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

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

2. Standard Equations and Graphs

The opening direction of a hyperbola is determined by the **positive term**. Regardless of the focus location, the following relationship always holds:

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

**Note**: This is the biggest difference from ellipses! In a hyperbola, $c$ (semi-focal length) is the hypotenuse of the right triangle and is the longest segment.

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

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

| **Graph** | | (Opens Up/Down) |

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

| **How to Identify** | **$x^2$ term is Positive** | **$y^2$ term is Positive** |

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

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

| **Asymptotes** | $y = \pm \frac{b}{a}x$ | $y = \pm \frac{a}{b}x$ |

3. Core Geometric Properties

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

1. **Range**: $|x| \ge a$ (The curve lies outside the vertices). The range of $y$ is all real numbers.

2. **Asymptotes**:

* This is the "skeleton" of the hyperbola. The curve approaches these lines infinitely but never touches them.

* **Calculation Trick**: Set the $1$ on the right side of the standard equation to $0$, i.e., $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 0$, and solve for $y = \pm \frac{b}{a}x$.

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

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

* Larger $e$ means a wider opening (flatter); smaller $e$ means a narrower opening (curvier).

4. Examples

**Example 1**: The foci are $(\pm 5, 0)$, and for a point $P$ on the curve, $||PF_1| - |PF_2|| = 6$. Find the equation and asymptotes.

**Solution**:

From definition, $2a = 6 \Rightarrow a = 3$.

From foci, $c = 5$.

Calculate $b^2$: In a hyperbola, $b^2 = c^2 - a^2 = 25 - 9 = 16$, so $b=4$.

Foci on x-axis, so equation is $\frac{x^2}{9} - \frac{y^2}{16} = 1$.

Asymptotes are $y = \pm \frac{4}{3}x$.

**Example 2**: Find the eccentricity and asymptotes of $4y^2 - 9x^2 = 36$.

**Solution**:

1. **Standard Form**: Divide by 36: $\frac{y^2}{9} - \frac{x^2}{4} = 1$.

2. **Identify Direction**: $y^2$ term is positive, so foci are on the **y-axis**.

3. **Parameters**: $a^2 = 9 \Rightarrow a = 3$ (Note: $a$ corresponds to the positive denominator), $b^2 = 4 \Rightarrow b = 2$.

4. **Calculate c**: $c = \sqrt{a^2 + b^2} = \sqrt{9+4} = \sqrt{13}$.

5. **Conclusion**:

* Eccentricity $e = \frac{c}{a} = \frac{\sqrt{13}}{3}$.

* Asymptotes: Set $\frac{y^2}{9} - \frac{x^2}{4} = 0 \Rightarrow y = \pm \frac{3}{2}x$.

5. Study Tips

* **Equation Recognition**: A "minus sign" means Hyperbola. The denominator of the positive term is $a^2$ (regardless of magnitude, unlike ellipses).

* **Asymptote Memory**: Don't rote memorize. Visualise a rectangle (length $2a$, width $2b$); the diagonals are the asymptotes.

* **Avoid Confusion**:

* **Hyperbola**: $c^2 = a^2 + b^2$ ($c$ is biggest)

* **Ellipse**: $a^2 = b^2 + c^2$ ($a$ is biggest)