Conic Form to Standard Form Calculator
Change any conic equation into a readable standard form. Review classification, center, rotation, and vertices. Build cleaner math reports with reliable steps every day.
Change any conic equation into a readable standard form. Review classification, center, rotation, and vertices. Build cleaner math reports with reliable steps every day.
Enter coefficients from Ax² + Bxy + Cy² + Dx + Ey + F = 0.
General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
Discriminant: B² - 4AC
Rotation angle: θ = 1/2 atan2(B, A - C)
Center equations: 2Ah + Bk + D = 0 and Bh + 2Ck + E = 0
If B² - 4AC is less than zero, the conic belongs to the ellipse family. If it equals zero, the conic belongs to the parabola family. If it is greater than zero, the conic belongs to the hyperbola family. Degenerate cases can still occur, so the calculator checks shifted constants and denominators.
| A | B | C | D | E | F | Type | Expected Standard Form |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 1 | -4 | 6 | -12 | Circle | (x - 2)² + (y + 3)² = 25 |
| 4 | 0 | 9 | -8 | 18 | -23 | Ellipse | (x - 1)² / 9 + (y + 1)² / 4 = 1 |
| 1 | 0 | -1 | -2 | -4 | -4 | Hyperbola | (x - 1)² - (y + 2)² = 1 |
| 1 | 0 | 0 | -4 | -8 | 12 | Parabola | (x - 2)² = 8(y - 1) |
A conic equation can look hard when every term is mixed together. This calculator turns that equation into a cleaner standard form. It reads six coefficients from the general equation. Then it checks the shape, rotation, center, vertex, and scale. The result is easier to graph and explain.
Standard form shows the useful features of a conic. A circle shows its center and radius. An ellipse shows its center and axis lengths. A hyperbola shows its center and transverse direction. A parabola shows its vertex and opening direction. These details are harder to see in general form.
Some conics include an xy term. That term usually means the graph is rotated. The calculator uses a rotation angle to remove the xy term. It then uses the rotated axes to form a simpler equation. This is useful for advanced algebra, analytic geometry, and engineering work.
Non-rotated conics often become standard form by completing the square. The x terms are grouped together. The y terms are grouped together. Then constants are moved to the other side. This process reveals shifts from the origin. The shifts become the center or vertex.
The calculator uses the discriminant B² - 4AC. A negative value points to an ellipse family. A zero value points to a parabola family. A positive value points to a hyperbola family. The tool also checks special cases. Some equations are points, lines, or have no real graph.
Students can use the calculator to verify homework steps. Teachers can create examples faster. Designers can inspect curves used in drawings. Physics users can study paths and mirrors. Data workers can document fitted conic equations. The export buttons make records simple.
Decimal answers depend on the chosen precision. Rounded values may look slightly different from exact symbolic work. For simple integer examples, the output is usually very clean. For rotated conics, decimal rotation values are normal. Always compare the result with your original equation when exact proof is required.
It is Ax² + Bxy + Cy² + Dx + Ey + F = 0. This single equation can represent circles, ellipses, parabolas, hyperbolas, and some degenerate graphs.
Standard form rewrites the equation so the main graph features are visible. It often shows the center, vertex, radius, axis lengths, or opening direction.
The xy term usually means the conic is rotated. The calculator finds a rotation angle and uses new axes to remove the mixed term.
It uses B² - 4AC. Negative values suggest ellipses. Zero suggests parabolas. Positive values suggest hyperbolas. Extra checks handle degenerate results.
Yes. Enter the circle equation coefficients. The result can show the center and radius squared in standard circle form.
Yes. When B is not zero, the calculator applies axis rotation. It then returns the standard equation in rotated U,V coordinates.
Enter zero. For example, if there is no xy term, use B = 0. If there is no y term, use E = 0.
A degenerate conic is a special case. It may become a point, a line pair, one line, or no real graph.
U and V are rotated coordinates. They appear when the original equation has an xy term and needs axis rotation.
The center solves two equations: 2Ah + Bk + D = 0 and Bh + 2Ck + E = 0. These come from partial derivatives.
Yes. It lists the classification method, rotation step, center or vertex process, and final standard form method.
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a clean report-style download.
Rotated conics often create decimal values for angles and coefficients. You can increase decimal precision for more detail.
Yes. Standard form gives the key graph data. You can use the center, vertex, axis lengths, and orientation to sketch the conic.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.