1. What is the Ellipse Function?

The ellipse function describes the set of all points (x, y) in a plane such that the sum of their distances from two fixed points (the foci) is constant. The standard equation of an ellipse centered at (h, k) is:

2. How Does the Calculator Work?

The calculator evaluates whether a given point (x, y) satisfies the ellipse equation:

Where:

• \( (x, y) \) — Point coordinates to test
• \( (h, k) \) — Center of the ellipse
• \( a \) — Semi-major axis length
• \( b \) — Semi-minor axis length

Interpretation:

• Result = 1: Point lies ON the ellipse
• Result < 1: Point lies INSIDE the ellipse
• Result > 1: Point lies OUTSIDE the ellipse

3. Importance of Ellipse Calculation

Applications: Ellipse calculations are crucial in astronomy (planetary orbits), engineering (elliptical gears), architecture (elliptical arches), and physics (particle motion).

4. Using the Calculator

Tips: Enter all values as real numbers. Semi-axes (a and b) must be positive. The calculator will determine if your point lies on, inside, or outside the defined ellipse.

5. Frequently Asked Questions (FAQ)

Q1: What if a = b?
A: When semi-major and semi-minor axes are equal (a = b), the ellipse becomes a circle.

Q2: How are the foci related to a and b?
A: For an ellipse, the distance of each focus from the center is \( c = \sqrt{a^2 - b^2} \) (assuming a > b).

Q3: What's the difference between major and minor axes?
A: The major axis (length 2a) is the longest diameter, while the minor axis (length 2b) is the shortest diameter.

Q4: Can the ellipse be oriented at an angle?
A: This calculator handles axis-aligned ellipses. Rotated ellipses require a more complex equation.

Q5: What are practical uses of this calculation?
A: Useful in collision detection (games/graphics), satellite orbit planning, and optical lens design.