Ellipse Properties and Equations
Master the key components, equations, and properties of ellipses in standard form.
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Questions Covered in This Set
10 cards to master
What is the definition of an ellipse?
The set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant.
What is the relationship between a, b, and c in an ellipse?
c² = a² - b², where a is the semi-major axis, b is the semi-minor axis, and c is the distance from center to each focus.
How do you determine if an ellipse has a horizontal or vertical major axis from its equation?
The larger denominator corresponds to a². If it's under the x-term, the major axis is horizontal; if under the y-term, it's vertical.
What is the standard form equation for an ellipse with a horizontal major axis?
(x-h)²/a² + (y-k)²/b² = 1, where a > b, with center at (h, k).
Where are the foci located in an ellipse with a vertical major axis centered at (h, k)?
At (h, k ± c), where c = √(a² - b²).
What does eccentricity measure in an ellipse?
Eccentricity e = c/a measures how 'stretched' or elongated an ellipse is, with 0 < e < 1.
What are the vertices of an ellipse?
The endpoints of the major axis, located at the farthest points from the center.
For the equation (x-2)²/25 + (y+1)²/9 = 1, what is the center and what are the values of a and b?
Center: (2, -1); a = 5 (since a² = 25); b = 3 (since b² = 9).
What steps are needed to convert general form to standard form?
Group x and y terms, complete the square for both variables, and divide through to get 1 on the right side.
What is the eccentricity of a circle?
0, because c = 0 in a circle (the foci coincide at the center).
