Problem 32 - Algebra and Trigonometry

Question

\(29-32\) . (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 equation represents a parabola, confirmed by the discriminant (\(\Delta = 0\)) and graphing.

Step 1: Identify the coefficients

We start with the given equation: \(9x^2 - 6xy + y^2 + 6x - 2y = 0\). In the standard conic section form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the coefficients are \(A = 9\), \(B = -6\), \(C = 1\), and \(D = 6\), \(E = -2\), \(F = 0\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) for conic sections is given by \(\Delta = B^2 - 4AC\). Substitute the values: \(\Delta = (-6)^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0\).

Step 3: Determine the conic based on the discriminant

If \(\Delta = 0\), the conic is a parabola. This tells us that the given equation forms a parabola.

Step 4: Graph the conic

To confirm the result, graph the equation \(9x^2 - 6xy + y^2 + 6x - 2y = 0\) using a graphing device or software. The graph should appear as a parabola, verifying the discriminant calculation.

Key Concepts

Discriminant

The discriminant is a valuable tool in mathematics for identifying the type of conic section represented by a quadratic equation. The general form of a quadratic equation in two variables is given by:\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\]The discriminant \(\Delta\) for conic sections is calculated using the formula:\[\Delta = B^2 - 4AC\]This formula helps us determine the nature of the 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 \(B = 0\)).

In our exercise, the equation provided is \(9x^2 - 6xy + y^2 + 6x - 2y = 0\). By substituting the coefficients \(A = 9\), \(B = -6\), and \(C = 1\) into the discriminant formula, we find \(\Delta = 0\). This indicates the equation describes a parabola.

Parabola

A parabola is a unique conic section that appears as a symmetrical, U-shaped curve on the graph. It has several important characteristics:

• The vertex, which is the highest or lowest point of the curve.
• A directrix, a line parallel to the axis of symmetry of the parabola.
• A focus, a point located such that every point on the parabola is equidistant from the focus and the directrix.

Parabolas can open either vertically or horizontally. A vertical parabola has the equation \(y = ax^2 + bx + c\), and a horizontal parabola can be expressed as \(x = ay^2 + by + c\). The direction in which the parabola opens depends on the coefficient of the squared term. In our exercise, substituting and solving for the dependent variable may reveal the orientation and vertex of the parabola, helping to visualize it graphically.

Graphing Conics

Graphing conics is an effective way to visualize the behavior of conic sections like parabolas, ellipses, and hyperbolas. Graphing these equations helps confirm theoretical computations, such as those made using the discriminant.To graph a conic section like a parabola, start by rearranging its equation into a recognizable form, if possible. Then, use graphing software or a graphing calculator. Here’s a step-by-step approach:

• Simplify the equation, if needed, to easily find key points like the vertex or intercepts.
• Plot important features such as the vertex and axis of symmetry for parabolas, or centers and foci for other conics.
• Draw the curve ensuring symmetry as specified by the conic type.

In the case of the equation \(9x^2 - 6xy + y^2 + 6x - 2y = 0\), plotting using a graphing device confirms the presence of a parabola. This graphical representation helps solidify the test's theoretical results, showing how mathematical calculations translate into visual forms.