Geometry: Understanding Ellipses

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Questions and Answers

What is the focal distance if the major axis and minor axis are 26 units and 10 units respectively?

10 units

16 units

14 units

12 units (correct)

What is the equation form of an ellipse with a center at the origin and foci at (0,8) and (0,-8)?

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

$\frac{y^2}{b^2} + \frac{x^2}{a^2} = 1$

$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$

$\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1$ (correct)

What is defined as the distance between the center and the focus of an ellipse?

Focal distance (correct)

Focal length

Vertex distance

Focal radius

Which of the following correctly represents the standard form of the equation for an ellipse centered at (h, k)?

$(x - h)^{2}/a^{2} + (y - k)^{2}/b^{2} = 1$ Signup and view all the answers

Using the values of a and c, how is b calculated?

$a = b + c$ Signup and view all the answers

What is the line that passes through the vertices and foci of an ellipse called?

Principal axis Signup and view all the answers

What is the focal distance of the ellipse represented by the equation $\frac{x^{2}}{36} + \frac{y^{2}}{100} = 1$?

8 units Signup and view all the answers

Based on the information given, if the major axis is vertical and measures 26 units, what can be inferred about the values of a and b?

a is greater than b Signup and view all the answers

If the major axis of an ellipse is vertical, how should the vertices be plotted relative to the center?

Up and Down Signup and view all the answers

Which equation represents an ellipse based on the equation $\frac{x^{2}}{36} + \frac{y^{2}}{49} = 1$?

Major axis is vertical. Signup and view all the answers

If an ellipse has vertices at 0, ±10 and co-vertices at ±8,0, what is the equation in standard form?

$\frac{x^2}{100} + \frac{y^2}{64} = 1$ Signup and view all the answers

Which of the following points is NOT crucial for graphing an ellipse?

Directrix Signup and view all the answers

If the center of an ellipse is located at (−1, 1), what is relevant when determining the layout of the axes?

The center affects the distance values. Signup and view all the answers

When converting the equation $3x + 4y - 24 = 0$ into standard form, what is the resulting format?

$\frac{x}{8} + \frac{y}{6} = 1$ Signup and view all the answers

Which of the following statements about the distance between two vertices is true?

It represents the length of the major axis. Signup and view all the answers

What is the general form of the equation after converting $\frac{x^{2}}{12} + \frac{y^{2}}{6} = 1$?

$6x + 12y - 72 = 0$ Signup and view all the answers

Study Notes

Ellipse Definition

An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci (F1 and F2), is constant.

Key Terms

Focal Distance: The distance between the center of the ellipse and a focus.
Major Axis: The line passing through the vertices, foci, and center of the ellipse. The length of the major axis is equal to the sum of the distances from a point on the ellipse to the two foci.
Vertices: The two points on the ellipse that lie on the major axis.
Minor Axis: A line segment perpendicular to the major axis, connecting the co-vertices of the ellipse.

Ellipse Equations

Standard Form (Center at Origin):
- x²/a² + y²/b² = 1 (Horizontal Major Axis)
- x²/b² + y²/a² = 1 (Vertical Major Axis)
Standard Form (Center at (h,k)):
- (x-h)²/a² + (y-k)²/b² = 1 (Horizontal Major Axis)
- (x-h)²/b² + (y-k)²/a² = 1 (Vertical Major Axis)

Key Relationships

Foci and Major/Minor Axes: In any ellipse, the distance between the two foci (2c) is related to the lengths of the major axis (2a) and minor axis (2b) by the equation: a² = b² + c²

Graphing Ellipses

Finding Key Points: To graph an ellipse, you need to find the center, foci, vertices, and co-vertices.
Orientation:
- If a > b, the major axis is horizontal. Vertices are plotted left and right of the center.
- If b > a, the major axis is vertical. Vertices are plotted above and below the center.
Co-vertices: Co-vertices are located on the minor axis and are plotted above and below the center (for horizontal major axis) or left and right of the center (for vertical major axis).

Converting Equations

Standard Form to General Form: Rearrange the standard-form equation to eliminate the fractions and put it in the form Ax² + By² + Cx + Dy + E = 0
General Form to Standard Form: Complete the square for both x and y terms to get the equation in standard form.

Example:

Finding an Ellipse Equation: Given that the center of the ellipse is at the origin, the major axis is vertical, and the major and minor axes are 26 units and 10 units respectively:
- a = 13 (half of the major axis length)
- b = 5 (half of the minor axis length)
- The equation is x²/25 + y²/169 = 1

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Description

This quiz explores the definition and properties of ellipses, including key terms like focal distance and axes. Test your understanding of the standard forms of ellipse equations and their relationships. Perfect for geometry enthusiasts!