Identify the conic represented by the equation without completing the square. 4x^2 - 9y^2 - 8x + 12y - 144 = 0

Verified step by step guidance

1. Identify the general form of the conic section equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).
2. Compare the given equation \( 4x^2 - 9y^2 - 8x + 12y - 144 = 0 \) with the general form to identify the coefficients: \( A = 4 \), \( B = 0 \), \( C = -9 \), \( D = -8 \), \( E = 12 \), \( F = -144 \).
3. Determine the type of conic by evaluating the discriminant \( B^2 - 4AC \). For this equation, calculate \( 0^2 - 4(4)(-9) \).
4. Since \( B^2 - 4AC > 0 \), the conic is a hyperbola.
5. Conclude that the equation \( 4x^2 - 9y^2 - 8x + 12y - 144 = 0 \) represents a hyperbola.

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The four primary types are circles, ellipses, parabolas, and hyperbolas. Each type has a distinct equation and geometric properties, which can be identified by analyzing the coefficients of the quadratic terms in the equation.

Standard Form of Conic Equations

Conic sections can be expressed in standard forms, which help in identifying their type. For example, the standard form for a hyperbola is (x-h)²/a² - (y-k)²/b² = 1, while for an ellipse it is (x-h)²/a² + (y-k)²/b² = 1. Recognizing the structure of the equation allows for quick identification of the conic type without completing the square.

Discriminant of Conic Sections

The discriminant of a conic section, given by the formula D = B² - 4AC from the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0, helps determine the type of conic. If D < 0, it represents an ellipse; D = 0 indicates a parabola; and D > 0 signifies a hyperbola. This method provides a straightforward way to classify conics based on their coefficients.