Standard equations of an ellipse

Put the centre at the origin with the foci F₁(−c, 0) and F₂(c, 0). Writing PF₁ + PF₂ = 2a with the distance formula, isolating one root, squaring, simplifying and squaring again leads to x²/a² + y²/(a² − c²) = 1, that is x²/a² + y²/b² = 1.

Conversely, a point on x²/a² + y²/b² = 1 has PF₁ = a + (c/a)x and PF₂ = a − (c/a)x, which add to 2a, so the equation describes exactly the ellipse.

With the foci on the y-axis the equation is x²/b² + y²/a² = 1. These two are the standard equations of an ellipse; ellipses with other centres or tilted axes are outside this chapter.

From x²/a² ≤ 1, −a ≤ x ≤ a and similarly −b ≤ y ≤ b: the ellipse sits inside the rectangle formed by x = ±a and y = ±b and touches its four sides.

If (x, y) is on the ellipse, so are (−x, y), (x, −y) and (−x, −y): an ellipse is symmetric about both coordinate axes.

Both foci sit on the major axis, and the bigger of the two denominators tells which axis that is: under x² it is the x-axis, under y² the y-axis.

x²/25 + y²/9 = 1: a = 5, b = 3, c = √(25 − 9) = 4. Foci (±4, 0), vertices (±5, 0), major axis 10, minor axis 6, e = 4/5.

9x² + 4y² = 36 divided by 36 is x²/4 + y²/9 = 1. Now y² has the larger denominator, so a = 3, b = 2 and c = √5: foci (0, ±√5), vertices (0, ±3), major axis 6, minor axis 4, e = √5/3.