Hyperbola Conic Sections Examples with Graphs

📚 Quick Study Guide: Hyperbolas Unveiled

• ✨ Definition: A hyperbola is the set of all points in a plane such that the absolute difference of the distances from two fixed points (called the foci) is constant.
• ↔️ Standard Form (Horizontal Transverse Axis): When the transverse axis is parallel to the x-axis, the equation is $ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $.
• ↕️ Standard Form (Vertical Transverse Axis): When the transverse axis is parallel to the y-axis, the equation is $ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 $.
• 📍 Center: The point $(h, k)$ is the center of the hyperbola.
• 🔵 Vertices: These are the endpoints of the transverse axis. For a horizontal hyperbola, they are $(h \pm a, k)$. For a vertical hyperbola, they are $(h, k \pm a)$.
• 🔥 Foci: These are the two fixed points that define the hyperbola. For a horizontal hyperbola, they are $(h \pm c, k)$. For a vertical hyperbola, they are $(h, k \pm c)$. The relationship between $a, b,$ and $c$ is $c^2 = a^2 + b^2$.
• 📏 Transverse Axis: The line segment connecting the two vertices. Its length is $2a$.
• ➖ Conjugate Axis: The line segment perpendicular to the transverse axis, passing through the center. Its length is $2b$.
• ↗️ Asymptotes: These are two lines that the hyperbola approaches but never touches as it extends infinitely. They pass through the center $(h, k)$.
  • 📝 For horizontal: $ y-k = \pm \frac{b}{a}(x-h) $
  • ✍️ For vertical: $ y-k = \pm \frac{a}{b}(x-h) $
• 🌟 Eccentricity: Denoted by $e = \frac{c}{a}$, for a hyperbola, $e > 1$. It describes how 'open' the hyperbola is.
• 📊 Graphing Tip: A helpful step is to sketch a central rectangle with sides $2a$ and $2b$ centered at $(h, k)$. The asymptotes pass through the corners of this rectangle. The vertices are on the transverse axis, passing through the midpoints of the relevant sides of the rectangle.

🧠 Practice Quiz: Test Your Hyperbola Knowledge

1. Which of the following is a key defining property of a hyperbola?

1. The sum of distances from two fixed points is constant.
2. The absolute difference of distances from two fixed points is constant.
3. The ratio of distances from a fixed point and a fixed line is equal to one.
4. It is the set of all points equidistant from a fixed point.

2. What is the center of the hyperbola given by the equation $ \frac{(x+3)^2}{16} - \frac{(y-2)^2}{9} = 1 $?

1. $(3, -2)$
2. $(-3, 2)$
3. $(3, 2)$
4. $(-3, -2)$

3. For the hyperbola $ \frac{y^2}{25} - \frac{x^2}{144} = 1 $, what are the coordinates of its vertices?

1. $(\pm 5, 0)$
2. $(0, \pm 12)$
3. $(\pm 12, 0)$
4. $(0, \pm 5)$

4. Given a hyperbola with $a=3$ and $b=4$, what is the value of $c$?

1. $3$
2. $4$
3. $5$
4. $7$

5. Which of the following equations represents the asymptotes for the hyperbola $ \frac{x^2}{36} - \frac{y^2}{49} = 1 $?

1. $ y = \pm \frac{7}{6}x $
2. $ y = \pm \frac{6}{7}x $
3. $ y = \pm \frac{x}{7} $
4. $ y = \pm \frac{x}{6} $

6. A hyperbola has a transverse axis of length 10 and a conjugate axis of length 24. What are the lengths of $a$ and $b$ respectively?

1. $a=10, b=24$
2. $a=5, b=12$
3. $a=12, b=5$
4. $a=20, b=48$

7. Consider a hyperbola centered at the origin with its foci on the y-axis. Which standard form would describe this hyperbola?

1. $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $
2. $ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 $
3. $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $
4. $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 $