The Ellipse
An ellipse is the locus of points whose distances from two foci sum to a constant, with standard form x2/a2 + y2/b2 = 1 and eccentricity e < 1.
What is The Ellipse?
An ellipse is the locus of points whose distances from two foci sum to a constant, with standard form x2/a2 + y2/b2 = 1 and eccentricity e < 1.
Key formula / rule: b2 = a2 (1 - e2) with a the semi-major axis.
Key points
- Compute the eccentricity, foci and axes of an ellipse.
- Derive the focal-distance property of the ellipse.
Common exam trap
Assuming a is always the denominator under x2.
Definitions
- Term
The Ellipse
- Meaning
An ellipse is the locus of points whose distances from two foci sum to a constant, with standard form x2/a2 + y2/b2 = 1 and eccentricity e < 1.
- Term
The Ellipse — explanation
- Meaning
The relation b2 = a2(1 - e2) ties the axes to the eccentricity, so any one of the three determines the others. Whether the major axis is horizontal depends on which denominator is larger.
Learning objectives
Compute the eccentricity, foci and axes of an ellipse.
Derive the focal-distance property of the ellipse.
Formulae
- Key point
b2 = a2 (1 - e2) with a the semi-major axis.
- Key point
Foci are at (+/- ae, 0) for a horizontal major axis.
- Key point
Sum of focal distances equals 2a for every point on the ellipse.
Prerequisites
MAT-U3-CS-T1-S1-C1
Common mistakes
Assuming a is always the denominator under x2.
Using the parabola eccentricity value of 1 in ellipse formulae.
Keywords
Ellipse
Practice preview
What is the standard equation of an ellipse centered at the origin with major axis along the x-axis?…
easy
An ellipse has a semi-major axis of length 5 and a semi-minor axis of length 3. What is its eccentricity?…
easy
For the ellipse given by the equation x^2/16 + y^2/9 = 1, what is the length of its major axis?…
easy