Problem 24 - Precalculus with Calculus Previews

Question

In Problems \(23-28,\) use the discriminant to identify the conic without actually graphing. $$ 2 x^{2}-2 x y+2 y^{2}=1 $$

Step-by-Step Solution

Answer

The conic is an ellipse.

Step 1: Identify Quadratic Form

First, let's identify the standard form of the given quadratic equation. The given equation is:\[ 2x^2 - 2xy + 2y^2 = 1 \]In the standard quadratic form for conic sections, it is given as:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Here, \( A = 2 \), \( B = -2 \), \( C = 2 \), \( D = 0 \), \( E = 0 \), and \( F = -1 \).

Step 2: Calculate the Discriminant

The discriminant for conic sections is calculated using the formula:\[ \Delta = B^2 - 4AC \]Substitute the values from the quadratic equation:\[ \Delta = (-2)^2 - 4(2)(2) \]\[ \Delta = 4 - 16 = -12 \]

Step 3: Determine the Type of Conic

The value of the discriminant \( \Delta \) tells us the type of conic:- If \( \Delta > 0 \), the conic is a hyperbola.- If \( \Delta = 0 \), the conic is a parabola.- If \( \Delta < 0 \), the conic is an ellipse (or a circle if \( A = C \) and \( B = 0 \)).Since \( \Delta = -12 < 0 \), the conic is an ellipse.

Key Concepts

Discriminant

The discriminant is a valuable tool in mathematics, particularly when dealing with conic sections. It helps us to identify the type of conic section represented by a given quadratic equation without having to graph it.
The discriminant for conic sections is derived from the general quadratic form and is calculated using the formula \( \Delta = B^2 - 4AC \). This formula stems from the coefficients of the quadratic equation, where \( A \), \( B \), and \( C \) are extracted.
Key points to remember about the discriminant in the context of conic sections:

• If \( \Delta > 0 \), it indicates a hyperbola.
• If \( \Delta = 0 \), it signifies a parabola.
• If \( \Delta < 0 \), the resulting shape is an ellipse or, in special cases, a circle.

Using the discriminant allows us to quickly assess the nature of the conic section in question, making it an efficient approach in algebra and geometry.

Ellipse

An ellipse is one of the four types of conic sections and is often defined by its oval shape. It is symmetric about its two principal axes.
Ellipses can be described in algebraic terms as an equation of the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \).
When dealing with the discriminant:

• An ellipse is characterized by a discriminant \( \Delta < 0 \).
• If \( A = C \) and \( B = 0 \), the conic is a circle, which is a special type of ellipse.

The geometric properties of an ellipse include its axes; the longest diameter is called the major axis, and the shortest one is the minor axis. The points where these axes intersect the ellipse are its vertices and co-vertices. Understanding ellipses helps in various scientific fields, including astronomy and physics, where orbital paths frequently appear as ellipses.

Quadratic Forms

Quadratic forms are algebraic expressions involving squared terms. They are generally written as \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \). This standard form is essential when working with conics.
In the context of conic sections, the coefficients \( A \), \( B \), and \( C \) determine the shape and alignment of the conic.
To dissect a quadratic form:

• Match each term with corresponding values like \( A \), \( B \), and \( C \).
• Utilize these coefficients to calculate the discriminant and ascertain the conic type.
• Observe that each type of conic section (ellipse, parabola, hyperbola) arises from different combinations of these coefficients.

Learning quadratic forms and their application in conic sections cultivates a deeper understanding of geometric curves.It provides a vital link between algebraic expressions and geometric interpretation, enhancing both problem-solving and visualization skills.