Problem 32 - Precalculus: Mathematics for Calculus

Question

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$9 x^{2}-6 x y+y^{2}+6 x-2 y=0$$

Step-by-Step Solution

Answer

The conic is a parabola, confirmed by a discriminant of zero and graphing.

Step 1: Identify the coefficients

The given equation is \(9x^2 - 6xy + y^2 + 6x - 2y = 0\). We need to identify the coefficients for the second-degree terms. In standard form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), we have: \(A = 9\), \(B = -6\), \(C = 1\).

Step 2: Calculate the discriminant

The discriminant of a conic is calculated using the formula \(B^2 - 4AC\). Substitute in the values of \(A\), \(B\), and \(C\): \[ (-6)^2 - 4(9)(1) = 36 - 36 = 0 \]

Step 3: Analyze the discriminant

The discriminant \(B^2 - 4AC\) helps determine the type of conic:- If \(B^2 - 4AC > 0\), the conic is a hyperbola.- If \(B^2 - 4AC = 0\), the conic is a parabola.- If \(B^2 - 4AC < 0\), the conic is an ellipse (or a circle if \(A = C\)).Here, since \(B^2 - 4AC = 0\), the conic is a parabola.

Step 4: Graph the conic

Graphing the given equation using a graphing device will confirm its shape as a parabola. Set up the device to plot the equation \(9x^2 - 6xy + y^2 + 6x - 2y = 0\) and observe the resultant graph. The graph should appear as a parabola, supporting our conclusion from the discriminant analysis.

Key Concepts

Discriminant

The concept of a discriminant is essential when analyzing conic sections. In mathematics, the discriminant helps us identify the specific type of conic section represented by a given quadratic equation. A quadratic equation can be expressed in the standard form: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \] The discriminant for a conic section is calculated using the formula \( B^2 - 4AC \). This formula gives us a simple number called the discriminant, which provides crucial information about the nature of the graph.

• If \( B^2 - 4AC > 0 \), the conic is a hyperbola.
• If \( B^2 - 4AC = 0 \), the conic is a parabola.
• If \( B^2 - 4AC < 0 \), it indicates the conic is an ellipse. It could also be a circle if additionally \( A = C \).

Knowing how to calculate and interpret the discriminant is crucial for anyone learning about conics as it simplifies identifying conic sections quickly.

Parabola

A parabola is a unique and interesting conic section. It can appear in various contexts, from the path of a projectile to the design of satellite dishes. In the context of conic sections, a parabola is characterized by its U-shaped curve. The above discriminant analysis showed that the equation we are dealing with represents a parabola because the discriminant is zero. This indicates the equation form is neither opening outward like a hyperbola nor enclosed like an ellipse or circle. Its general equation can be expressed when the discriminant \( B^2 - 4AC = 0 \). Parabolas have several important properties:

• They have a directrix and a focus, which helps define their shape.
• The vertex is the point where the parabola changes direction, located midway between the focus and the directrix.
• Parabolas are symmetric about their axis of symmetry, which passes through the vertex.

Understanding parabolas is an integral part of mastering conic sections, providing insights into larger geometric and algebraic concepts.

Graphing Conics

Graphing conics is a practical tool in visualizing the shape and properties of conic sections. Due to the complexity of their equations, using graphing devices or graphing calculators can be particularly helpful. When graphing the respective equation \(9x^2 - 6xy + y^2 + 6x - 2y = 0\), you will notice the shape that confirms our discriminant analysis — a parabola. Here are simple steps to graph a conic:

• Simplify the equation to the standard form, if necessary, to identify the coefficients \(A\), \(B\), and \(C\).
• Compute the discriminant \(B^2 - 4AC\) to determine the type of conic section.
• Use graphing tools to plot the equation. Visual representation helps confirm and better understand the nature of your conic section.

Graphing accurately portrays the symmetry and defining features of the conic, enhancing comprehension of both simple and complex conics.