Rule for Classifying Conics Using Discriminant

Given the equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

Compute: \( \Delta = B^2 - 4AC \)

? Based on value of \( \Delta \):

• Ellipse: \( \Delta < 0 \) and \( A \ne C \), \( B \ne 0 \) → tilted ellipse
• Circle: \( \Delta < 0 \) and \( A = C \), \( B = 0 \)
• Parabola: \( \Delta = 0 \)
• Hyperbola: \( \Delta > 0 \)

Example:

For the equation: \( 3x^2 + 10xy + 11y^2 + 14x + 12y + 5 = 0 \)

\( A = 3 \), \( B = 10 \), \( C = 11 \) →
\( \Delta = 10^2 - 4(3)(11) = 100 - 132 = -32 \)

Since \( \Delta < 0 \), it represents an ellipse.