Problem 61 - Algebra 2

Question

Which of the following is an equation of a parabola? A. \(2 x^{2}-5 y^{2}+4 x-36=0\) B. \(2 x^{2}-5 y+4 x-36=0\) C. \(2 x^{2}+2 y^{2}+4 x-12 y-36=0\) D. \(2 x^{2}+5 y^{2}-36=0\)

Step-by-Step Solution

Answer

The right answer is D. \(2 x^{2}+5 y^{2}-36=0\) is the equation of a parabola.

Step 1: Analyzing Option A

This equation is of the form \(Ax^{2} + By^{2} + Cx + D = 0\), which represents the equation of an ellipse, not a parabola.

Step 2: Analyzing Option B

This equation is of the form \(Ax^{2} + Cy + Dx + E = 0\), which is neither a straight line, circle, parabola, ellipse nor hyperbola.

Step 3: Analyzing Option C

This equation is of the form \(Ax^{2} + By^{2} + Cx + Dy + E = 0\), which is the equation of a circle or an ellipse, not a parabola.

Step 4: Analyzing Option D

This equation is of the form \(Ax^{2} + By^{2} + C = 0\), which is the equation of a parabola.

Key Concepts

Quadratic Equations

Quadratic equations are expressions of the second degree, involving terms with variables raised to the power of two. They are generally written in the form \(ax^2 + bx + c = 0\). In this form, 'a', 'b', and 'c' are coefficients where \(a eq 0\). The solutions of quadratic equations, also known as roots, can be found using various methods such as factoring, completing the square, or using the quadratic formula:

• The quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Factoring: This involves expressing the quadratic equation as a product of its linear factors.
• Completing the square: This transforms the equation into a perfect square trinomial, making it easier to solve.

The graph of a quadratic equation is a parabola, a U-shaped curve that can open either upwards or downwards depending on the sign of the leading coefficient 'a'. If 'a' is positive, the parabola opens upwards; if negative, it opens downwards. Quadratic equations are essential in modeling various real-world phenomena, from projectile motion to optimization problems.

Conics

Conic sections, or conics, are curves obtained by intersecting a plane with a double-napped cone. The main types of conics include parabolas, ellipses, and hyperbolas. The general equation representing conic sections is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Whether the equation represents a parabola, ellipse, or hyperbola depends on the values of the coefficients.

• Parabolas: Represented by equations where either \(A = 0\) or \(C = 0\) if there is no squared term in one of the variables. Parabolas are U-shaped and have a single axis of symmetry.
• Ellipses: Characterized by \(A\) and \(C\) having the same sign and not equal zero. They are oval-shaped and include circles as a special case when \(A = C\).
• Hyperbolas: Occur when \(A\) and \(C\) have opposite signs, resulting in two symmetrical open branches.

Recognizing the type of conic by analyzing the coefficients and structure of the equation is a crucial skill in algebra and geometry. This helps in solving real-world problems where modeling is required.

Analyzing Equations

Analyzing equations involves understanding the form and structure of an equation to identify the type of curve it represents. This includes:

• Identifying the highest degree of each term: This determines the family of the curve.
• Examining the coefficients: These tell us about the shape and orientation of the curve.
• Checking the presence of mixed terms: Terms like \(Bxy\) indicate a rotation of the conic section.

In the example provided, we explored four equations:

• Option A: Recognizable as an ellipse due to both \(x^2\) and \(y^2\) terms having non-zero coefficients that are both positive or negative.
• Option B: This form doesn't fit any specific conic classification, lacking a \(y^2\) term.
• Option C: Suggestive of a circle or ellipse, as both squared terms are present, indicating symmetry in all directions.
• Option D: Identified as a parabola since only the \(x^2\) term is present, aligning with common parabola equations.

Understanding the underlying structure helps in solving problems and visualizing the geometric representation of the equation.