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Chapter 6: Problem 53

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x^{2}+3 y^{2}+8 x-24 y+51=0$$

Short Answer

Expert verified
The given equation is the equation of an ellipse.

Step by step solution

01

Rewriting as Complete Squares

First, we will rewrite the given equation such that we complete the square for terms involving \(x\) and \(y\). To do so, we group the terms involving like variables. In this case, we rearrange the equation to become \(4x^2 + 8x + 3y^2 - 24y = -51\). Notice that we can factor out 4 from the \(x\)-terms and 3 from the \(y\)-terms, resulting in \(4(x^2 + 2x) + 3(y^2 - 8y)= -51\). We then add squares of half of the coefficients of \(x\) and \(y\), respectively inside parentheses: \(4[(x+1)^2 -1] + 3[(y - 4)^2 -16] = -51\). Simplifying this, we get \(4(x+1)^2 + 3(y - 4)^2 = -51+4+48\).
02

Transforming into Standard Form

Next, we transform the equation into its standard form. Simplifying the right-hand side, we get \(4(x+1)^2 + 3(y - 4)^2 = 1\). This looks like the standard form but we need to get in the form \((x-h)^2/a^2 + (y - k)^2/b^2 = 1\). We finally get \((x+1)^2/(0.25) + (y - 4)^2/(1/3) = 1\) by dividing both sides of the equation by 1.
03

Identifying the Conic Section

Our equation \((x+1)^2/(0.25) + (y - 4)^2/(1/3) = 1\), is now in the standard form of an ellipse because \(a^2\) and \(b^2\) are both positive and are not equal. So, the equation represents an ellipse.

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Most popular questions from this chapter

\(r=3 \sin 2 \theta\)

Consider the polar equation $$ r=\frac{4}{1-0.4 \cos \theta} $$ (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. $$ r_{1}=\frac{4}{1+0.4 \cos \theta}, \quad r_{2}=\frac{4}{1-0.4 \sin \theta} $$ (c) Use a graphing utility to verify your results in part (b).

The equation $$ r=\frac{e p}{1 \pm e \sin \theta} $$ is the equation of an ellipse with \(e<1\). What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

\(r=3-4 \cos \theta\)

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{lll} {\text { Conic }} & \text { Eccentricity } & \text { Directrix } \\ \text { Hyperbola } & e=2 & x=1 \\ \end{array} $$
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