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Chapter 6: Problem 55

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

Short Answer

Expert verified
The graph of the equation represents a hyperbola.

Step by step solution

01

Rewrite the equation in a more readable format

Rewrite the given equation as \(4x^{2} - 4x - y^{2} = 3\).
02

Complete the square of the quadratic part

Rearrange and complete the squares for the variables, which becomes \(4(x^{2} - x) - y^{2} = 3\). Add and subtract \(\frac{1}{4}\) inside the parentheses. So the equation becomes \(4[(x - \frac{1}{2})^{2} - \frac{1}{4}] - y^{2} = 3\). Simplify the equation to \((2x - 1)^{2} - y^{2} = 4\).
03

Recognize the equation as a standard form of a conic section

Now, from the equation \(a(x - h)^{2} - b(y - k)^{2} = 1 \), the solution found resembles the form of the standard equation for a hyperbola. Therefore, the graph represents a hyperbola.

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Most popular questions from this chapter

In Exercises \(129-132,\) determine whether the statement is true or false. Justify your answer. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\).

A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveyor. (c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor.

Consider a line with slope \(m\) and \(y\) -intercept \((0,4)\) (a) Write the distance \(d\) between the point \((3,1)\) and the line as a function of \(m\) (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the point and the line. (d) Is it possible for the distance to be \(0 ?\) If so, what is the slope of the line that yields a distance of \(0 ?\) (e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem.

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=-3 \sin \theta$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=\pi / 6$$
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