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Ellipse Center Formula:
for the general ellipse equation:
\[ ax² + cy² + bx + dy + e = 0 \]
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The center of an ellipse is the midpoint of both the major and minor axes. It's the point of symmetry for the ellipse and serves as a reference point for its equation in standard form.
The calculator uses the formula:
Derived from the general ellipse equation: \[ ax² + cy² + bx + dy + e = 0 \]
Explanation: The center coordinates (h, k) are found by completing the square for both x and y terms in the general ellipse equation.
Details: Knowing the center is essential for graphing the ellipse, converting between standard and general forms, and analyzing its geometric properties.
Tips: Enter the coefficients a, b, c, and d from the general ellipse equation. Coefficients a and c must be non-zero (they determine the ellipse's orientation).
Q1: What if a or c is zero?
A: The equation would not represent an ellipse (it would be a parabola or degenerate case). The calculator requires non-zero values for a and c.
Q2: How is this different from circle center?
A: For circles (a special case of ellipse where a = c), the formula simplifies to the same form but with equal denominators.
Q3: What units are used for the center?
A: The center coordinates are in the same units as the original equation's variables (x and y).
Q4: Can I use this for rotated ellipses?
A: This formula works only for ellipses whose axes are parallel to the coordinate axes. Rotated ellipses require additional calculations.
Q5: How precise are the results?
A: Results are rounded to 4 decimal places. For exact fractions, symbolic computation would be needed.