Pre-Calculus Test Questions on Ellipse Definition by Foci

📚 Quick Study Guide

• 📍 Definition: An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant.
• 📐 Standard Equation: The standard form equation of an ellipse centered at $(h, k)$ depends on whether the major axis is horizontal or vertical:
  • Horizontal Major Axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $a > b$
  • Vertical Major Axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, where $a > b$
• 🔥 Foci: The foci are located along the major axis, $c$ units from the center. The relationship between $a$, $b$, and $c$ is $c^2 = a^2 - b^2$.
• 📏 Major and Minor Axes: The major axis has a length of $2a$, and the minor axis has a length of $2b$.
• 🎯 Vertices: The vertices are the endpoints of the major axis.

Practice Quiz

1. Which of the following statements best describes the definition of an ellipse based on its foci?

   1. The set of all points where the difference of the distances from two fixed points is constant.
   2. The set of all points where the product of the distances from two fixed points is constant.
   3. The set of all points where the sum of the distances from two fixed points is constant.
   4. The set of all points equidistant from two fixed points.

2. Given an ellipse with foci at $(-3, 0)$ and $(3, 0)$, and the sum of the distances from any point on the ellipse to the foci is 10, what is the value of 'a' in the standard equation?

   1. 3
   2. 5
   3. 6
   4. 10

3. An ellipse has foci at $(0, -4)$ and $(0, 4)$. If a point on the ellipse is 6 units from one focus and 14 units from the other, what is the length of the major axis?

   1. 8
   2. 10
   3. 16
   4. 20

4. The foci of an ellipse are at $(1, 2)$ and $(7, 2)$. The sum of the distances from any point on the ellipse to the foci is 10. What is the center of the ellipse?

   1. (4, 2)
   2. (1, 2)
   3. (7, 2)
   4. (4, 4)

5. For an ellipse, the distance between the foci is 8, and the sum of the distances from any point on the ellipse to the foci is 12. What is the value of $b^2$ in the equation of the ellipse?

   1. 4
   2. 9
   3. 20
   4. 36

6. The equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$ represents an ellipse. What are the coordinates of the foci?

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

7. Which of the following equations represents 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. $\frac{x^2}{36} + \frac{y^2}{20} = 1$
   4. $\frac{x^2}{20} + \frac{y^2}{36} = 1$