Examples of Polar Equations

[Technology Exerc...

Question

Examples of Polar Equations

[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.

r = −2 cos θ

Accepted Answer

Welcome back everyone. Draw the graph of R equals 4 sin theta. For this problem, let's go ahead and multiply both sides by r, so we are going to get R2 equals 4 r sin theta. And now we're going to use the definition of R2, which is X2 + y2, and polar coordinates, and we also have to remember that the Y coordinate is equal to r sin theta. While the x coordinate is r cosine theta. So in this case, we get x2 + y2 on the left hand side. On the right hand side we have 4 multiplied by r sine theta, which is 4 y. We can now subtract 4 y from both sides to get x2 + y2 minus 4 y equals 0. And now we will complete the square. In order to do that, we want to add 4 to both sides of the equation, so we're going to add 4 on the left and 4 on the right. Now we can complete the square to get x 2 plus y minus 4 divided by 2 gives us 2, right? squared equals 4, and this is the equation of the circle with a center x noty not at 0, right, x minus 022, because we have y minus 2 squad, right, and essentially 2 is the Y coordinate of the center. Now, what is the radius? Remember that the radius is the square root of the right hand side, so that's square root of 4, which is 2. So what we want to do is add a center, so that's 0.2. Let's add that. 0.2 and we can plot a circle with that center and a radius of 2. So we can basically add four points we can go 2 units to the right from the center horizontally so that would be. X equals 2, Y equals 2. Let's specifically add that point. We can also go to the left horizontally, so that would be -2.2. And now vertically we can go up along the y axis. Buy 2 units to get 0.4, and we can go down to get the origin. Now we can basically connect those 4 points in a circle, so let's go ahead and do that. And optimize our sketch to ensure that the coordinates are consistent. Well then, so we have our final graph, which is the equation of the circle with a center at 0.2 and the radius of 2. Thank you for watching.

Examples of Polar Equations[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.r = −2 cos θ

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Examples of Polar Equations

[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.

r = −2 cos θ