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Chapter 6: Problem 49

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$y^{2}-4 x^{2}+4 x-2 y-4=0$$

Short Answer

Expert verified
The graph of the equation \(y^{2}-4 x^{2}+4 x-2 y-4=0\) represents a hyperbola.

Step by step solution

01

Simplify the Equation

The first step is to rearrange and simplify the equation, grouping together terms containing \(x\) and \(y\). This can be done as follows: \(-4 x^{2} + 4 x + y^{2} - 2y = 4 \). This can be rewritten as: \(-4 (x^{2} - x) + (y^{2} - 2y) = 4 \)
02

Rearrange into conic standard form

Next we need to express the equation in the standard form of each conic section. For an ellipse \(\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1\) and for a hyperbola \(\frac{(x-h)^{2}}{a^{2}} - \frac{(y-k)^{2}}{b^{2}} = 1\) or vice versa. To this end, we complete the square for \(x\) and \(y\) terms separately. So, \(-4 [(x-0.5)^{2} - 0.25] + (y - 1)^{2} - 1 = 4 \) and simplified we have \((y - 1)^{2} - 4 (x - 0.5)^{2} = 5 \). Dividing by 5 to compare with the standard forms, we get: \(\frac{(y - 1)^{2}}{5} - \frac{(x - 0.5)^{2}}{1.25} = 1.\)
03

Identify the conic section

In the resulting equation, the \(y^{2}\) term is positive and the \(x^{2}\) term is negative, and they are subtracted from each other. This corresponds to the standard form of a hyperbola with a vertical axis of symmetry. Thus, the graph of the equation represents a hyperbola.

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Most popular questions from this chapter

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Parabola } & (5, \pi) \\ \end{array} $$

\(r=\frac{6}{2 \sin \theta-3 \cos \theta}\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (4, \pi / 2),(1, \pi / 2) \end{array} $$

Consider the equation \(r=3 \sin k \theta\). (a) Use a graphing utility to graph the equation for \(k=1.5\). Find the interval for \(\theta\) over which the graph is traced only once. (b) Use a graphing utility to graph the equation for \(k=2.5\). Find the interval for \(\theta\) over which the graph is traced only once. (c) Is it possible to find an interval for \(\theta\) over which the graph is traced only once for any rational number \(k\) ? Explain.

In Exercises 29-32, use a graphing utility to graph the rotated conic. $$ r=\frac{2}{1-\cos (\theta-\pi / 4)} $$
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