 A quadratic curve it is a curve representing a given quadratic equation. The curve y = ax2 + bx + c, with, is called the quadratic function of a variable. In general, the curve f (x, y) = Ax2 + Bxy + Cxy2 + Dx + Ey + F is called quadratic function; the sign of the discriminant «B2 - 4AC ”of said function determines the shape of the curve: parabola, ellipse or circle, or hyperbola.

## Curve type identification

1. Expand and simplify the given quadratic equation.
2. Rearrange the given equation and rewrite it as the general formula “Ax2 + Bxy + Cxy2 + Dx + Ey + F = 0 ”.
3. Compare the equation obtained in step 2 with the general formula to find y.
4. Finding the discriminant «B2 - 4AC ”:
• If B2 - 4AC = 0, then the given equation represents a Parabola.
• If B2 - 4AC <0, then the given equation represents a ellipse or a circle.
• If B2 - 4AC> 0, then the given equation represents a hyperbola.

Example 1: Identify the type of curve for the equation y2 - 4y - 2x + 3 = 0.

Rearranging the equation in the general formula “Ax2 + Bxy + Cxy2 + Dx + Ey + F = 0 ”:

0x2 - 0xy + (1) (1) and2 - 2x - 4y + 3 = 0

Comparing both equations we determine that:

A = 0; B = 0; C = 1; D = - 2; E = - 4 and F = 3.

Now, when evaluating the discriminant we obtain that:

B2 - 4AC = (0)2 - 4 (0) (1) = 0

Therefore, the equation y2 - 4y - 2x + 3 = 0 represents a parabola.

Example 2: Identify the type of curve for the 7x equation2 +5 + 2xy + 4y2 + 3y - 22 = 0.

Rearranging the equation in the general formula “Ax2 + Bxy + Cxy2 + Dx + Ey + F = 0 ”:

7x2 + 2xy + (4) (1) and2 - 5x + 3y - 22 = 0.

Comparing both equations we determine that:

A = 7; B = 2; C = 4; D = - 5; E = 3 and F = - 22.

Now, when evaluating the discriminant we obtain that:

B2 - 4AC = (7)2 - 4 (3) (4)> 0

Therefore, the 7x equation2 +5 + 2xy + 4y2 + 3y - 22 = 0. represents either an ellipse or a circle.

Example 3: Identify the type of curve for the 8x equation2 + 3x + 4y2 + 12y = 0.

Rearranging the equation in the general formula “Ax2 + Bxy + Cxy2 + Dx + Ey + F = 0 ”:

8x2 - 0xy + (4) (1) and2 + 3x + 12y + 0 = 0.

Comparing both equations we determine that:

A = 8; B = 0; C = 4; D = 3; E = 12 and F = 0.

Now, when evaluating the discriminant we obtain that:

B2 - 4AC = (0)2 - 4 (8) (4) <0

Therefore, the 8x equation2 + 3x + 4y2 + 12y = 0 represents a hyperbola.