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Chapter 6: Problem 56

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 y^{2}+4 x^{2}-24 x+35=0$$

Short Answer

Expert verified
The graph of the equation \(4y^{2}+4x^{2}-24x+35=0\) is a circle with center at (3, 0) and radius 0.5.

Step by step solution

01

Simplify the given equation

In the given equation \(4y^2 + 4x^2 - 24x + 35 = 0\), we can simplify the equation by dividing every term by 4 to make the coefficients of \(x^2\) and \(y^2\) equal to 1, which will result in: \(y^2 + x^2 - 6x + 8.75 = 0\)
02

Complete the square on x

We will complete the square on x by adding the square of half the coefficient of x to both sides of the equation. Here, the coefficient of x is -6, so half of it will be -3 and its square will be 9. The resulting equation will be: \(y^2 + (x - 3)^2 + 8.75 - 9 = 0\) which simplifies to \(y^2 + (x - 3)^2 - 0.25 = 0\)
03

Identify the graph

We have now simplified the equation to the form \(x^2 + y^2 - h^2 = 0\), which is in the standard form of a circle's equation \((x-h)^2 + (y-k)^2 = r^2\), where (h,k) is the center of the circle and r is its radius. Therefore, the equation represents a circle with center at (3, 0) and radius 0.5.

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Most popular questions from this chapter

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.

In Exercises 33-48, find a polar equation of the conic with its focus at the pole. $$ \begin{array}{ll} {\text { Conic }} & \text { Vertex or Vertices } \\ \text { Hyperbola } & (2,0),(8,0) \\ \end{array} $$

\(r=4 \cos \theta\)

In Exercises 73-78, solve the trigonometric equation. $$ \sqrt{2} \sec \theta=2 \csc \frac{\pi}{4} $$

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