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Chapter 8: Problem 20

Convert the rectangular equation to polar form and sketch its graph. $$ x^{2}+y^{2}-2 a x=0 $$

Short Answer

Expert verified
The polar form of the given rectangular equation is \(r(r-2a \cdot cos(胃)) = 0\), and its graph includes a single point at the origin and a circle centered at (a,0) with a radius a.

Step by step solution

01

Substitute in the values of x and y

Use the formulas \(x = r \cdot cos(胃)\), \(y = r \cdot sin(胃)\) and substitute in the provided equation. The rectangular equation \(x^{2} + y^{2} - 2ax = 0\) turns into \((r \cdot cos(胃))^{2} + (r \cdot sin(胃))^{2} - 2a \cdot r \cdot cos(胃) = 0\).
02

Simplify the equation

Now, simplify this equation by applying the identities. The identity \(sin^2(胃) + cos^2(胃) = 1\) is helpful in this context. The above equation simplifies to \(r^{2}-2a \cdot r \cdot cos(胃) = 0\).
03

Factoring the simplified equation

Factor out \(r\) from the equation to arrive at the final polar equation. The factored equation will become \(r(r-2a \cdot cos(胃)) = 0\).
04

Sketching the graph

This is the equation of a circle in polar coordinates. The impact made by the \(r = 0\) term is just a single point at the origin. The other factor \(r-2a \cdot cos(胃) = 0\) can be rewritten as \(r = 2a \cdot cos(胃)\) which is a circle centered at (a,0) with radius a. The graph of this equation will consist of this circle together with the point at the origin.

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Most popular questions from this chapter

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{-1}{1-\cos \theta}\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 \cos \theta, \quad y=6 \sin \theta $$

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Folium of Descartes: } x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Describe what happens to the distance between the directrix and the center of an ellipse if the foci remain fixed and \(e\) approaches 0 .

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^{2}\) and \(y=t^{2}\) is the line \(y=x\).
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