Ellipse Standard Form Calculator

Convert general ellipse equation to standard form by completing the square. Find center, axes, vertices, and foci.

General Form: Ax2 + Cy2 + Dx + Ey + F = 0

Enter the coefficients (B=0 for axis-aligned ellipse)

Standard Form

Standard Form Equation
--
Center (h, k)--
Semi-major axis (a)--
Semi-minor axis (b)--
Major axis direction--
Vertices (major)--
Co-vertices (minor)--
Foci--
Eccentricity (e)--
Area--

Step-by-Step: Completing the Square

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

What Is the Standard Form of an Ellipse?

The standard form of an ellipse centered at (h, k) is:

(x - h)2 / a2 + (y - k)2 / b2 = 1

where a is the semi-major axis and b is the semi-minor axis. If a > b, the major axis is horizontal. If b > a in the denominators (meaning the larger denominator is under y), the major axis is vertical.

How to Convert: Completing the Square

Step 1: Group Terms

Group the x-terms and y-terms together, and move the constant to the other side.

A(x^2 + D/A*x) + C(y^2 + E/C*y) = -F

Step 2: Complete the Square

For each group, add (coefficient/2)^2 inside and compensate on the right side.

A(x + D/2A)^2 + C(y + E/2C)^2 = K

Step 3: Divide by K

Divide both sides by K to get 1 on the right. Read off center, a, and b.

(x-h)^2/(K/A) + (y-k)^2/(K/C) = 1

Conditions for a Valid Ellipse

  • Both A and C must be positive (or both negative, in which case multiply through by -1).
  • A and C must be different (if A = C, it is a circle, not an ellipse).
  • The constant K on the right side after completing the square must be positive (otherwise no real ellipse exists).
  • This calculator handles axis-aligned ellipses (no Bxy cross term). For rotated ellipses, a rotation of axes would be needed first.

Finding Vertices and Foci

Once you have the standard form, the key features are straightforward to find:

  1. Center: (h, k) directly from the equation.
  2. Semi-axes: a = sqrt of the larger denominator, b = sqrt of the smaller denominator.
  3. Vertices: Along the major axis at distance a from center.
  4. Co-vertices: Along the minor axis at distance b from center.
  5. Foci: Along the major axis at distance c = sqrt(a2 - b2) from center.
  6. Eccentricity: e = c / a.

Example Walkthrough

Convert 4x2 + 9y2 - 16x + 54y + 29 = 0 to standard form:

  1. Group: 4(x2 - 4x) + 9(y2 + 6y) = -29
  2. Complete squares: 4(x2 - 4x + 4) + 9(y2 + 6y + 9) = -29 + 16 + 81 = 68
  3. Factor: 4(x - 2)2 + 9(y + 3)2 = 68
  4. Divide: (x - 2)2/17 + (y + 3)2/(68/9) = 1
  5. Center: (2, -3), a2 = 17, b2 = 68/9

Common Mistakes

  • Forgetting to factor out the coefficient before completing the square.
  • Not adding the same value to both sides when completing the square.
  • Confusing which axis is the major axis (it corresponds to the larger denominator).
  • Not checking whether the equation actually represents an ellipse (A and C must have the same sign and be unequal).

Special Cases

  • Circle: If A = C, the equation represents a circle, not an ellipse. The "semi-axes" are both equal to the radius.
  • Centered at origin: If D = 0 and E = 0, the ellipse is centered at (0, 0) and the equation is already nearly in standard form.
  • Degenerate: If the right side K equals 0, the "ellipse" degenerates to a single point. If K is negative, there is no real curve.