Calculator
Use the general circle form x² + y² + Dx + Ey + F = 0. Decimals and simple fractions are accepted for D, E, and F.
Example Data Table
| General Equation | Center | Radius | Standard Form | Type |
|---|---|---|---|---|
| x² + y² - 6x + 8y + 9 = 0 | (3, -4) | 4 | (x - 3)² + (y + 4)² = 16 | Real circle |
| x² + y² + 4x - 10y + 13 = 0 | (-2, 5) | 4 | (x + 2)² + (y - 5)² = 16 | Real circle |
| x² + y² - 2x - 2y - 7 = 0 | (1, 1) | 3 | (x - 1)² + (y - 1)² = 9 | Real circle |
| x² + y² + 2x + 6y + 10 = 0 | (-1, -3) | 0 | (x + 1)² + (y + 3)² = 0 | Point circle |
Formula Used
The calculator starts with the general circle equation:
Completing the square is applied to the x-terms and y-terms separately.
y2 + Ey = (y + E/2)2 - (E/2)2
After rearrangement, the standard circle form becomes:
From the coefficients, the geometric values are:
- Center: h = -D/2 and k = -E/2
- Radius squared: r² = (D² + E²)/4 - F
- Radius: r = √r² when r² is nonnegative
- Diameter: 2r
- Circumference: 2πr
- Area: πr²
If r² is positive, the equation represents a real circle. If r² is zero, the graph collapses to one point. If r² is negative, the equation has no real circle on the coordinate plane.
How to Use This Calculator
- Write your equation in the form x² + y² + Dx + Ey + F = 0.
- Enter the values of D, E, and F in the three coefficient fields.
- Choose the decimal precision you want for the output.
- Adjust plot points and graph padding if you want a denser or wider graph.
- Press the calculate button to display the result section above the form.
- Read the transformed standard form, center, radius, and derived measurements.
- Check the step-by-step algebra to verify each completed square.
- Use the CSV or PDF buttons to save the result summary.
FAQs
1) What does this calculator solve?
It converts a circle equation from general form into standard form using completing the square. It also gives the center, radius, area, circumference, intercepts, domain, range, and graph.
2) Why is completing the square useful for circles?
Completing the square reveals the center and radius directly. That makes the geometry easier to interpret, graph, and verify than leaving the equation in its expanded form.
3) What happens when the radius squared is negative?
A negative radius squared means the equation does not produce a real circle on the coordinate plane. The calculator labels it as an imaginary circle and skips the real-circle geometry.
4) Can this handle zero values for D or E?
Yes. If either coefficient is zero, the center lies directly on one axis. The same completing-square process still works, and the calculator shows the correct standard form.
5) Does the calculator show intercepts?
Yes. It solves the related quadratic equations for x-intercepts and y-intercepts. If no real intercepts exist, the result clearly states that outcome.
6) What is a point circle?
A point circle occurs when the radius equals zero. The equation still has a center, but every point on the circle collapses into that single location.
7) Can I enter fractions?
Yes. The coefficient fields accept decimals and simple fractions such as 3/2, -7/4, or 5. The calculator converts them before performing the algebra.
8) What does the graph show?
The graph plots the circle boundary and marks the center. For a point circle, it shows the single point. For an imaginary circle, no real graph is drawn.