Conic General Form Converter | Naruhodo Tutoring

Equation Setup

x² + y² − 4x + 6y − 3 = 0

Enter coefficients from Ax² + Cy² + Dx + Ey + F = 0. (B=0; no xy term.)

A (x²)

C (y²)

D (x)

E (y)

Examples

Result

Enter coefficients above and press Identify & Convert to see the conic type, standard form, and graph.

Standard Form

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Key Properties

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Classification Rules (B = 0)

Condition	Conic Type	Discriminant B²−4AC
A = C ≠ 0	Circle	< 0
A ≠ C, same sign	Ellipse	< 0
A and C opposite signs	Hyperbola	> 0
A = 0 or C = 0 (not both)	Parabola	= 0
A = 0 and C = 0	Line / Degenerate	—

The discriminant B²−4AC < 0 for ellipses/circles, = 0 for parabolas, > 0 for hyperbolas. Since B = 0 here, this simplifies to −4AC.

Completing the Square

Ax² + Dx = A(x + D/2A)² − D²/4A

General idea: Group terms by variable, factor out leading coefficients, then add the square of the half-coefficient to both sides.

For x: Ax² + Dx = A[x² + (D/A)x] = A[(x + D/2A)² − (D/2A)²]

For y: Cy² + Ey = C[y² + (E/C)y] = C[(y + E/2C)² − (E/2C)²]

After completing both squares, move constants to the right side and divide to put 1 (or the conic form) on the right.

Always factor A out of x-terms before completing the square.
Whatever you add inside the parenthesis gets multiplied by A on the right too.
After dividing through, check that the equation matches the expected standard form.

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