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Chapter 10: Problem 28

(a) Use the discriminant to identify the conic. (b) Confirm your answer by graphing the conic using a graphing device. $$x^{2}-2 x y+3 y^{2}=8$$

Short Answer

Expert verified
The conic is an ellipse, confirmed by graphing.

Step by step solution

01

Identify the coefficients

Write down the general form of the conic section equation, which is \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). For the given equation \(x^2 - 2xy + 3y^2 = 8\), identify the coefficients: \(A = 1\), \(B = -2\), \(C = 3\), \(D = 0\), \(E = 0\), and \(F = -8\).
02

Calculate the discriminant

The discriminant formula for a conic is \(B^2 - 4AC\). Substitute the identified coefficients: \((-2)^2 - 4(1)(3) = 4 - 12 = -8\).
03

Analyze the discriminant

Determine the type of conic section using the discriminant. If \(B^2 - 4AC < 0\), it indicates an ellipse. Based on the discriminant \(-8\), which is less than zero, the given conic section is an ellipse.
04

Confirm by graphing

Use a graphing device to graph the equation \(x^2 - 2xy + 3y^2 = 8\). Observe that the graph resembles an ellipse, thus confirming our discrimination analysis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant in Conic Sections
In conic sections, the discriminant is a crucial tool for identifying the type of conic section represented by a quadratic equation. For an equation in the general form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant is given by the formula \(B^2 - 4AC\). This formula helps to determine the nature of the 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 \(A = C\) and \(B = 0\).
By using the discriminant, we confirmed that the equation \(x^2 - 2xy + 3y^2 = 8\) describes an ellipse, as the discriminant \(-8\) is less than zero.
Understanding what the discriminant indicates allows us to quickly and effectively identify the conic, aiding us when graphing it later on.
Understanding Ellipses
An ellipse is a type of conic section that looks like an elongated circle. The defining feature of an ellipse is that it has two main axes: the major axis and the minor axis.
  • The major axis is the longer diameter across the ellipse.
  • The minor axis is the shorter diameter across the ellipse.
  • The points where the axes meet are the center of the ellipse.
To understand ellipses better, imagine a circle stretched in one direction, which forms an oval shape. This shape can be described mathematically by its equation. An ellipse centered at the origin can be written as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively. Note that if \(a\) equals \(b\), the ellipse is actually a circle. Recognizing an ellipse from its equation helps with visualizing and graphing it correctly based on its properties.
Graphing Conic Sections
Graphing conic sections, like ellipses, requires an understanding of their equations and properties. With today鈥檚 technology, graphing devices can simplify this task considerably. When you input the equation \(x^2 - 2xy + 3y^2 = 8\) into a graphing calculator or software, it transforms the equation into a visual format.
Here are tips to graph conics effectively:
  • Understand the general shape the conic will take based on the discriminant.
  • Use the equation to find key features like vertices, axes, and center.
  • Ensure the graphing tool settings accommodate variations in the conic鈥檚 scale.
  • Analyze the plotted graph to identify if the shape matches expectations, confirming calculations.
By graphing the conic sections, students gain a deeper insight into how these shapes translate from equations to geometric forms. This visualization helps in confirming the conic type determined algebraically, closing the loop between theoretical math and practical application.

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Most popular questions from this chapter

Complete the square to determine whether the equation represents an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. $$x^{2}+4 y^{2}+20 x-40 y+300=0$$

(a) Use rotation of axes to show that the following equation represents a hyperbola: \(7 x^{2}+48 x y-7 y^{2}-200 x-150 y+600=0\) (b) Find the \(X Y\) - and \(x y\) -coordinates of the center, vertices, and foci. (c) Find the equations of the asymptotes in \(X Y\) - and \(x y\) -coordinates.

Two stones are dropped simultaneously in a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola. (IMAGE CAN'T COPY)

A polar equation is given. (a) Express the polar equation in parametric form. (b) Use a graphing device to graph the parametric equations you found in part (a). $$r=2^{\theta / 12}, \quad 0 \leq \theta \leq 4 \pi$$

This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus. (a) Draw graphs of the family of parabolas $$x^{2}=4 p(y+p)$$ for \(p=-2,-\frac{3}{2},-1,-\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\). (b) Show that each parabola in this family has its focus at the origin. (c) Describe the effect on the graph of moving the vertex closer to the origin.
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