Find the foci, vertices, eccentricity, and graph of any ellipse from its standard equation.
(x−h)²/a² + (y−k)²/b² = 1, a ≥ b > 0
An ellipse is the set of all points where the sum of distances to the two foci is constant and equal to 2a (twice the semi-major axis).
The center is at (h, k). The larger denominator tells you the direction of the major axis: larger under x → horizontal; larger under y → vertical.
Key relationship: a² = b² + c², so c = √(a² − b²). The foci always lie inside the ellipse along the major axis, each at distance c from the center.
e = c/a, 0 < e < 1
Eccentricity measures how "stretched" an ellipse is. It always falls between 0 and 1 for an ellipse:
e → 0: The foci merge at the center and the shape approaches a perfect circle.
e → 1: The foci move toward the ends of the major axis and the ellipse becomes very flat and elongated.
Planets orbit the Sun in ellipses. Earth's orbit has e ≈ 0.017 (nearly circular), while Halley's Comet has e ≈ 0.967 (very elongated).
One-on-one Algebra 2 tutoring builds real intuition for ellipses, hyperbolas, and parabolas — we work through your actual homework and tests so the patterns click.