Textbook Question
Identify the conic represented by the equation without completing the square. 4x^2 - 9y^2 - 8x + 12y - 144 = 0
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8. Conic Sections
Parabolas
2:53 minutesTextbook Question
Identify the conic represented by the equation without completing the square. y^2 + 4x + 2y - 15 = 0
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Rewrite the given equation in a more standard form by grouping terms involving the same variable: \( y^2 + 2y + 4x - 15 = 0 \).
Identify the highest degree of each variable. Notice that \( y^2 \) is the highest degree term, and there is no \( x^2 \) term. This suggests the equation might represent a parabola.
Recall the general form of a parabola: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). For a parabola, either \( A = 0 \) or \( C = 0 \). In this case, \( C = 1 \) (coefficient of \( y^2 \)) and \( A = 0 \) (no \( x^2 \) term), confirming it is a parabola.
The presence of the linear \( x \) term (\( 4x \)) and the linear \( y \) term (\( 2y \)) indicates that the parabola is not centered at the origin and may need further manipulation to find its vertex.
Conclude that the given equation represents a parabola. To fully analyze its properties (e.g., vertex, focus, etc.), you would complete the square, but this step is not required for this problem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 variables in the equation.
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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 parabola is y = ax^2 + bx + c, while that of a circle is (x-h)² + (y-k)² = r². By rearranging the given equation into a recognizable standard form, one can determine the specific conic represented.
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Discriminant of Conic Sections
The discriminant of a conic section, given by the formula D = B² - 4AC from the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, helps classify the conic. If D < 0, it represents an ellipse; if D = 0, a parabola; and if D > 0, a hyperbola. This classification is crucial for identifying the type of conic without completing the square.
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