Problem 27 - Algebra and Trigonometry

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$2 x^{2}-4 x y+2 y^{2}-5 x-5=0$$

Step-by-Step Solution

Answer

The conic is a parabola, confirmed by the discriminant \(\Delta = 0\) and graphing.

Step 1: Identify the coefficients of the conic equation

First, let's identify the coefficients of the given conic equation. The general form of a conic equation is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). Here, comparing with the given equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\), we find: \(A = 2\), \(B = -4\), \(C = 2\), \(D = -5\), \(E = 0\), and \(F = -5\).

Step 2: Calculate the Discriminant of the Conic Section

The discriminant \(\Delta\) of a conic section is given by the formula \(\Delta = B^2 - 4AC\). Substituting the coefficients found, we have: \(\Delta = (-4)^2 - 4 \cdot 2 \cdot 2\). Calculate this to get \(\Delta = 16 - 16 = 0\).

Step 3: Interpret the Discriminant Result

The value of the discriminant determines the type of conic section. If \(\Delta = 0\), the conic is a parabola. Since we've found \(\Delta = 0\) for this equation, the conic section is a parabola.

Step 4: Verify by Graphing

To confirm the classification, graph the equation using a graphing tool or calculator. Input the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\) and check the graph. A parabola will appear on the graph, confirming the discriminant's classification.

Key Concepts

Discriminant

The discriminant is a powerful tool in analyzing conic sections. It helps in identifying the type of conic represented by a quadratic equation in two variables. Every conic section can be characterized using a standard quadratic form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]Here, the discriminant \( \Delta \) is calculated as:\[ \Delta = B^2 - 4AC \]This discriminant aligns with the quadratic function characteristics to determine the conic section type:

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse. Additionally, if \( A = C \) and \( B = 0 \), it is a circle.

Knowing how to calculate and interpret the discriminant allows quick identification of the nature of a conic section.

Conic Section Identification

Identifying conic sections involves recognizing the nature of the graph or equation you are dealing with. Once the discriminant is determined, you can classify the conic section easily. For the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\), the coefficients are:

• \( A = 2 \)
• \( B = -4 \)
• \( C = 2 \)

The discriminant \( \Delta \) is computed as \( \Delta = B^2 - 4AC = (-4)^2 - 4 \times 2 \times 2 = 0 \). Since \( \Delta = 0 \), this classifies the equation as a parabola.
Understanding the coefficient relationships and their implications through the discriminant simplifies the identification process, saving time and minimizing errors.

Graphing Conic Sections

Graphing is another essential method to verify the conic section type shown by its equation. For more accuracy and intuition, one can use graphing calculators or software tools. When graphing the equation \(2x^2 - 4xy + 2y^2 - 5x - 5 = 0\), you should verify the following:- The shape on the graph should align with the type identified by the discriminant- For this specific case, you should see a parabola when plottingUsing graphing tools helps match the discriminant analysis with visual confirmation, ensuring that you correctly identify and understand the features and dimensions of the conic section.