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 Does the Calculator Work?

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

Where:

• \( h = -\frac{D}{2A} \) — x-coordinate of center
• \( k = -\frac{E}{2C} \) — y-coordinate of center
• \( a = \sqrt{\frac{-F + \frac{D^2}{4A} + \frac{E^2}{4C}}{A}} \) — semi-major axis
• \( b = \sqrt{\frac{-F + \frac{D^2}{4A} + \frac{E^2}{4C}}{C}} \) — semi-minor axis

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. Ensure x² and y² coefficients are positive and have the same sign for a real ellipse.

5. Frequently Asked Questions (FAQ)

Q1: What if I get negative denominators?
A: This means your equation doesn't represent a real ellipse (it might be imaginary or another conic section).

Q2: How do I know which is semi-major vs semi-minor?
A: The larger denominator corresponds to the semi-major axis (a), the smaller to semi-minor (b).

Q3: What if there's an xy term?
A: This calculator handles only non-rotated ellipses. An xy term indicates rotation and requires additional steps.

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

Q5: What units should I use?
A: Use consistent units for all coefficients. The calculator preserves your original units.