1. What is an Ellipse?

An ellipse is a closed curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. It's a stretched circle and one of the conic sections.

2. Understanding the Equation

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

Where:

• (h,k) — Center of the ellipse
• a — Length of semi-major axis
• b — Length of semi-minor axis
• If a > b, the major axis is horizontal
• If b > a, the major axis is vertical

3. Ellipse Properties

Key Properties:

• Major Axis: The longest diameter (length = 2a or 2b, whichever is larger)
• Minor Axis: The shortest diameter (length = 2a or 2b, whichever is smaller)
• Eccentricity: Measures how much the ellipse deviates from being circular (0 = circle, approaches 1 = very elongated)
• Area: π × a × b
• Circumference: No exact formula, but can be approximated

4. Using the Calculator

Instructions: Enter the center coordinates (h,k), semi-major axis (a), and semi-minor axis (b). The calculator will compute all key properties of the ellipse.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a circle and an ellipse?
A: A circle is a special case of an ellipse where a = b. All points are equidistant from the center in a circle.

Q2: How do you find the foci of an ellipse?
A: For a horizontal major axis: foci are at (h±c,k) where c² = a² - b². For vertical major axis: foci are at (h,k±c).

Q3: What are real-world applications of ellipses?
A: Planetary orbits, whispering galleries, satellite dishes, architectural designs, and medical equipment.

Q4: Can a and b be equal?
A: Yes, when a = b the equation becomes a circle with radius a.

Q5: What if my ellipse is not axis-aligned?
A: This calculator handles only standard axis-aligned ellipses. Rotated ellipses require more complex equations.