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

검증된 단계별 안내

Step 1: Begin by identifying the general form of the given equation. The equation is written as 4x^2 - 9y^2 - 8x + 12y - 144 = 0. This is a second-degree equation in two variables (x and y).

Step 2: Group the terms involving x and y separately. The equation can be rewritten as (4x^2 - 8x) + (-9y^2 + 12y) - 144 = 0.

Step 3: Observe the coefficients of the squared terms (4x^2 and -9y^2). Since the coefficients of x^2 and y^2 have opposite signs (one positive and one negative), this indicates that the conic is a hyperbola.

Step 4: Recall the standard form of a hyperbola: \( \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \) or \( \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \). The given equation can be transformed into one of these forms by completing the square, but this step is not required here to identify the conic.

Step 5: Conclude that the conic represented by the given equation is a hyperbola based on the signs of the squared terms and their coefficients.

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Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The four primary types of conic sections 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 identify their type. For example, the standard form of a circle is (x-h)² + (y-k)² = r², while a hyperbola is represented as (x-h)²/a² - (y-k)²/b² = 1. By rearranging the given equation into a recognizable standard form, one can determine the specific conic section it represents.

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 classify the conic. If D < 0, the conic is an ellipse (or circle); if D = 0, it is a parabola; and if D > 0, it is a hyperbola. This classification is crucial for identifying the type of conic represented by the equation.