Problem 3 - Contemporary Precalculus

Question

Assume that the graph of the equation is a nondegenerate conic section. Without graphing, determine whether the graph an ellipse, hyperbola, or parabola. $$x^{2}+2 x y+y^{2}+2 \sqrt{2} x-2 \sqrt{2} y=0$$

Step-by-Step Solution

Answer

Question: Without plotting the graph, determine whether the given equation represents an ellipse, hyperbola, or parabola: $$x^2 + 2xy + y^2 + 2\sqrt{2}x - 2\sqrt{2}y=0$$ Answer: The given equation represents a parabola.

Step 1: Write the Equation in General form

First, we'll write the given equation in general form. A general conic equation is written as: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ Comparing the given equation with the general conic equation, we have: $$x^2 + 2xy + y^2 + 2\sqrt{2}x - 2\sqrt{2}y=0$$ Here \(A=1\), \(B=2\), \(C=1\), \(D=2\sqrt{2}\), \(E=-2\sqrt{2}\), and \(F=0\).

Step 2: Determine the Discriminant

To determine the type of conic section, we'll calculate the discriminant, which is given by the formula: $$D=\; B^2 - 4AC$$ Plugging in the values, we have: $$D=\; (2)^2 - 4\times1\times1 = 4 - 4 = 0$$

Step 3: Identify the Conic Section

Now we'll use the discriminant to identify the type of conic section: 1. If \(D > 0\), it's a hyperbola 2. If \(D = 0\), it's a parabola 3. If \(D < 0\), it's an ellipse In this case, the discriminant, \(D\), is equal to zero. Therefore, the graph of the given equation represents a parabola.

Key Concepts

Parabola

The parabola is a unique shape among conic sections because it is defined as the set of points equidistant from a fixed point called the focus and a line called the directrix. This property leads to its characteristic U-shaped curve, which can open in any direction: up, down, left, or right. Parabolas are commonly encountered in everyday experiences, like the path of a thrown ball or the design of satellite dishes.

The equation of a parabola can vary depending on its orientation. For instance, a parabola opening upwards or downwards is typically expressed as \( y = ax^2 + bx + c \), where the vertex form is helpful in identifying its vertex easily. Meanwhile, one that opens to the left or right could be written as \( x = ay^2 + by + c \). The general form we see in conic sections will have certain coefficients that determine its precise structure and orientation—it's uniquely identified when the discriminant, as we'll discuss next, equals zero.

Discriminant in Conic Sections

The discriminant is a valuable tool for discerning the type of conic section an equation represents without needing to graph it. The discriminant is calculated with the formula \( D = B^2 - 4AC \), derived from the coefficients of the general conic equation: \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).

This clever mathematical expression helps categorize the conic sections:

• If \( D > 0 \), the equation represents a hyperbola.
• If \( D = 0 \), the expression is a parabola.
• If \( D < 0 \), the conic section is an ellipse.

In the given example, when calculating the discriminant of the equation \( x^2 + 2xy + y^2 + 2\sqrt{2}x - 2\sqrt{2}y = 0 \), it turned out to be zero, identifying it unequivocally as a parabola. This simplifies the recognition of the conic section, streamlining the problem-solving process.

General Form of Conic Sections

The general form of a conic section is an elegant and powerful representation that contains all the necessary information to identify and analyze any conic. This form is expressed as:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]where:

• \( A, B, \) and \( C \) are coefficients that determine the shape and type of the conic section.
• \( D \) and \( E \) pertain to the linear components and shift the graph in the coordinate plane.
• \( F \) is the constant term.

Understanding how these coefficients interact is crucial. For instance, when \( B^2 - 4AC = 0 \), the conic section is a parabola. When identifying a conic section from its general form, placing it into the context of the discriminant helps pinpoint its exact nature, enhancing comprehension of its geometric properties.