Test Your Knowledge: Writing the Equation of an Ellipse from Conditions

📚 Quick Study Guide

• 🔎 Standard Equation (Horizontal Major Axis): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis.
• 💡 Standard Equation (Vertical Major Axis): $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis. Note that $a > b$.
• 📝 Relationship between $a$, $b$, and $c$ (focal distance): $c^2 = a^2 - b^2$. This helps find the foci of the ellipse.
• 🧭 Center: The midpoint of the major axis.
• 📏 Vertices: The endpoints of the major axis. They are a distance of '$a$' from the center.
• 🎯 Foci: Located on the major axis, a distance of '$c$' from the center.
• 🧮 Eccentricity: A measure of how 'stretched' the ellipse is. $e = \frac{c}{a}$.

🧪 Practice Quiz

1. What is the standard form equation of an ellipse centered at (2, -3) with a horizontal major axis of length 10 and a minor axis of length 6?
   1. $\frac{(x-2)^2}{25} + \frac{(y+3)^2}{9} = 1$
   2. $\frac{(x+2)^2}{25} + \frac{(y-3)^2}{9} = 1$
   3. $\frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} = 1$
   4. $\frac{(x+2)^2}{9} + \frac{(y-3)^2}{25} = 1$
2. An ellipse has vertices at (5, 0) and (-5, 0) and foci at (3, 0) and (-3, 0). What is its equation?
   1. $\frac{x^2}{25} + \frac{y^2}{16} = 1$
   2. $\frac{x^2}{16} + \frac{y^2}{25} = 1$
   3. $\frac{x^2}{9} + \frac{y^2}{25} = 1$
   4. $\frac{x^2}{25} + \frac{y^2}{9} = 1$
3. The equation of an ellipse is $\frac{(x+1)^2}{4} + \frac{(y-2)^2}{9} = 1$. What are the coordinates of the center?
   1. (1, 2)
   2. (-1, 2)
   3. (1, -2)
   4. (-1, -2)
4. An ellipse has a center at (0, 0), a focus at (0, 4), and a vertex at (0, 5). What is the equation of the ellipse?
   1. $\frac{x^2}{9} + \frac{y^2}{25} = 1$
   2. $\frac{x^2}{25} + \frac{y^2}{9} = 1$
   3. $\frac{x^2}{16} + \frac{y^2}{25} = 1$
   4. $\frac{x^2}{25} + \frac{y^2}{16} = 1$
5. What is the equation of an ellipse with foci at (\pm 2, 0) and a major axis of length 6?
   1. $\frac{x^2}{9} + \frac{y^2}{5} = 1$
   2. $\frac{x^2}{5} + \frac{y^2}{9} = 1$
   3. $\frac{x^2}{36} + \frac{y^2}{4} = 1$
   4. $\frac{x^2}{4} + \frac{y^2}{36} = 1$
6. The endpoints of the major axis of an ellipse are (1, 6) and (1, -2), and the minor axis has length 4. Find the standard equation of the ellipse.
   1. $\frac{(x-1)^2}{4} + \frac{(y-2)^2}{16} = 1$
   2. $\frac{(x-1)^2}{16} + \frac{(y-2)^2}{4} = 1$
   3. $\frac{(x+1)^2}{4} + \frac{(y+2)^2}{16} = 1$
   4. $\frac{(x+1)^2}{16} + \frac{(y+2)^2}{4} = 1$
7. An ellipse has the equation $4x^2 + 9y^2 = 36$. Find the length of the major and minor axes.
   1. Major: 4, Minor: 6
   2. Major: 3, Minor: 2
   3. Major: 12, Minor: 8
   4. Major: 6, Minor: 4