1. What is the Standard Ellipse Equation?

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, and a and b are the semi-major and semi-minor axes lengths.

2. How Does the Calculator Work?

The calculator uses the standard ellipse equation:

Where:

• \( h \) — x-coordinate of center
• \( k \) — y-coordinate of center
• \( a \) — Length of semi-major axis
• \( b \) — Length of semi-minor axis

Explanation: The equation describes all points (x,y) that form an ellipse centered at (h,k) with specified axis lengths.

3. Importance of Ellipse Properties

Details: Understanding ellipse properties is crucial in mathematics, physics, engineering, and astronomy for describing orbital paths, optical systems, and various geometric applications.

4. Using the Calculator

Tips: Enter the center coordinates (h,k) and the lengths of semi-axes a and b. All values must be valid (a > 0, b > 0).

5. Frequently Asked Questions (FAQ)

Q1: What if a equals b?
A: If a = b, the ellipse becomes a circle with radius equal to a (or b).

Q2: How is eccentricity calculated?
A: Eccentricity e = √(1 - (b²/a²)) for a > b, or √(1 - (a²/b²)) for b > a. It measures how elongated the ellipse is.

Q3: What is the area of an ellipse?
A: Area = π × a × b, where a and b are the semi-major and semi-minor axes.

Q4: Can the ellipse be rotated?
A: The standard form assumes axes are parallel to coordinate axes. Rotated ellipses require a more general quadratic equation.

Q5: What are the foci of an ellipse?
A: For a horizontal ellipse (a > b), foci are at (h±c, k) where c² = a² - b². For vertical ellipse (b > a), foci are at (h, k±c) where c² = b² - a².