Problem 21 - Contemporary Precalculus

Question

Use the discriminant to identify the conic section whose equation is given, and find a viewing window that shows a complete graph. $$17 x^{2}-12 x y+8 y^{2}-80=0$$

Step-by-Step Solution

Answer

Based on the given information, the conic section is an ellipse, as the discriminant Δ is negative (-400). However, to determine a suitable viewing window to show the complete graph, additional information or further methods (e.g., completing the square) are needed to derive the canonical form of the ellipse. In the current form, it is not possible to strictly determine the viewing window.

Step 1: Determining the discriminant

First, let's find the discriminant Δ of the given equation. We have A = 17, B = -12, and C = 8. Applying the discriminant formula: $$Δ = (-12)^2 - 4(17)(8) = 144 - 544 = -400$$ The discriminant Δ is negative. Based on the discriminant values provided in the analysis, the given equation represents an ellipse.

Step 2: Finding the viewing window

To find the viewing window that shows the complete graph, we must first find the center of the graph. For an ellipse, the equation can be rewritten in the canonical form as follows: $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ To find the center (h, k) and the values of a and b, we need to rearrange the given equation in the required form. Divide both sides of the given equation by -80: $$\frac{17 x^{2}}{-80} -\frac{12 x y}{-80} + \frac{8 y^{2}}{-80} = 1$$ Now the equation can be written as: $$\frac{x^{2}}{\frac{80}{17}} - \frac{xy}{\frac{40}{3}} + \frac{y^{2}}{10} = 1$$ However, not every ellipse can be represented directly in the canonical form as we are missing the appropriate transformations. In this case, it is not possible to provide a fixed range for the viewing window that encompasses the entire ellipse while not knowing necessary information about the center, major and minor axes. To provide an approximate range, we can consider zooming out on a graph to include a reasonable range for x and y coordinates based on the coefficients. For instance, a range of (-10, 10) may work for both x and y-axis, but there is no guarantee that this range will fully represent the ellipse. It would require additional information about the ellipse's transformation or further methods to derive the canonical form, such as completing the square with respect to x and y. In the given form, it is not possible to strictly determine the viewing window.

Key Concepts

Discriminant

In the realm of conic sections, the discriminant is a crucial component that helps in determining the type of conic represented by a quadratic equation. For any equation of the form \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]the discriminant is calculated using the formula \[\Delta = B^2 - 4AC.\]Here's what the discriminant tells us:

• If \(\Delta > 0\), the conic is a hyperbola.
• If \(\Delta = 0\), it is a parabola.
• If \(\Delta < 0\), the conic is an ellipse (or a circle, which is a special case of an ellipse).

In our problem, the discriminant was calculated to be \(-400\), indicating that the conic section is an ellipse. This negative result is key for identifying the nature of the graph we wish to study.

Ellipse

An ellipse is a type of oval shape that results from slicing a cone with a plane at an angle. It is defined by two fixed points known as foci, and the sum of the distances from any point on the ellipse to each focus is constant.

The standard form equation of an ellipse centered at \((h, k)\) is \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1,\]where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

In cases where we deal with conics given by general quadratic equations, as in our exercise, identifying the precise parameters of an ellipse may require additional transformations like completing the square. Understanding the equation of an ellipse helps us in visualizing the graph and determining appropriate viewing windows.

Quadratic Equations

Quadratic equations are polynomials of degree two and take the form \(ax^2 + bx + c = 0\). When dealing with two variables, especially in the context of conic sections, they reveal much about various shapes like ellipses and hyperbolas.

In coordinate geometry, the generalized quadratic form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) represents various conics. The coefficients \(A, B,\) and \(C\) and their relationships govern the type of conic section.

While solving these equations, it's essential to consider transformations like completing the square or rotation of axes to explore distinct forms that better describe the geometry of the corresponding curves.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, merges algebra with geometry through the use of a coordinate system. This field deals with the study of geometric figures using algebraic equations. By applying algebraic methods to geometric problems, it becomes easier to resolve various properties and relationships between points, lines, and curves.

Conic sections are a significant part of coordinate geometry as they can be defined through the second-degree polynomial equations based on general quadratic forms.
By analyzing coefficients and discriminants in coordinate systems, complexities regarding the nature, orientation, and position of these shapes can be clarified and visualized. Working within coordinate geometry also allows for practical applications such as determining trajectories or optimizing spatial configurations. Thus, it's a powerful tool in further understanding the relationship between algebraic equations and geometrical figures.