Cartesian to Polar Equations

Find polar equat...

Question

Cartesian to Polar Equations

Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

x² + y² + 5y = 0

Accepted Answer

Welcome back everyone to another video. Find the polar equation for the circle x2 + y2 minus 8y equals 0. Also sketch its graph. For this problem, let's remember that x is equal to r multiplied by cosine theta. Y is equal to r multiplied by sine theta and also we know that x2 + y2 is equal to r2. So for the given equation, what we can do is simply replace x2 with r cosine theta to get r2 multiplied by cosine 2 theta. Plus y2 becomes r 2 sin 2 theta and essentially we can either use these and factor out r2 or we can basically show that x2 + y2 is equal to r2. So for the first sum we are just going to writer 2. In our equation, and we're subtracting 8 multiplied by y, which is 8 r sin theta, and this difference is equal to 0. Now we're going to factor outr which gives us r multiplied by r minus 8 sine theta is equal to 0. And we're going to solve for R. So we have two solutions. Either R is equal to 0, we're going to admit to this solution specifically. Or r minus 8 sin theta is equal to 0. So now we're going to solve for R. R is equal to 8 sine theta. And this is the equation that we wanted to get. Now the reason why r equals 0 is omitted is simply because it's the origin, right? Since the origin is included in the second equation, when theta is equal to 0 or pi, we can simplify the polar equation and simply use r equals a sin theta as the only solution. So this is our equation. And now we are going to sketch the graph. To sketch the graph of the circle, we're going to complete the square. We can write x2 + y2 minus 8y + 16. We're going to add 16 to both sides, so we get 16 on the right hand side as well. And now we can simply write it as x2 plus y minus 8 divided by 2 gives us +42 equals 16. So now this is a circle with a center of 0.4. Let's add that point, 0.4. So that's our center with a radius of R equals square root of 16, right? Because on the right-hand side we have the square of the radius, and that's for. So now, we are going to add our center, 0.4. Let's label the coordinates, 0.4. And because the radius is 4, we can basically go 4 units down along the y axis to get. Our first point for the circle, right, so that would be 0.0, which is basically the origin, then we can go horizontally to the right, which is going to give us 4.4. So we can also add that point 4.4. We can go to the left horizontal, which is going to give us -4. 4, and finally, we can go up along the y axis, which is going to give us 0.8. So we have those four points and we can basically connect them in a circle. Well then, so we have our final answers and thank you for watching.

Cartesian to Polar EquationsFind polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.x² + y² + 5y = 0

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Cartesian to Polar Equations

Find polar equations for the circles in Exercises 33–36. Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations.

x² + y² + 5y = 0