1. What is the Standard Form of an Ellipse?

The standard form of an ellipse equation is \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\), where (h,k) is the center, a is the semi-major axis, and b is the semi-minor axis. This form clearly shows all key geometric properties of the ellipse.

2. How the Conversion Works

The calculator converts the general quadratic form \(Ax^2 + Cy^2 + Dx + Ey + F = 0\) to standard form by:

Key Steps:

• Center calculation: \(h = -\frac{D}{2A}\), \(k = -\frac{E}{2C}\)
• Axis lengths: \(a = \sqrt{\frac{h^2A + k^2C - F}{A}}\), \(b = \sqrt{\frac{h^2A + k^2C - F}{C}}\)

3. Importance of Standard Form

Details: The standard form immediately reveals the ellipse's center, axes lengths, orientation, and eccentricity, making it essential for graphing and geometric analysis.

4. Using the Calculator

Tips: Enter all coefficients from your general ellipse equation. The calculator will complete the square and return the standard form with all geometric parameters.

5. Frequently Asked Questions (FAQ)

Q1: What if I get negative values under the square roots?
A: This indicates the equation doesn't represent a real ellipse (it might be imaginary or another conic section).

Q2: How do I know which axis is major?
A: The larger denominator indicates the major axis (a > b means horizontal major axis).

Q3: What if the equation includes xy terms?
A: This calculator handles only axis-aligned ellipses (B=0). Rotated ellipses require additional transformation.

Q4: Can I use this for circles?
A: Yes, circles are special ellipses where a = b (equal denominators).

Q5: How precise are the results?
A: Results are calculated to high precision but displayed with 3 decimal places for readability.