37–48. Polar-to-Cartesian coordinates Convert the following eq...

Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 6 cos θ + 8 sin θ

Accepted Answer

Welcome back, everyone. Convert the polar equation R equals -2 cosine theta plus 6 sin theta to Cartesian coordinates and describe the resulting curve. For this problem let's recall that. X equals R cosine theta and Y equals R sine theta. This is the relationship between Cartesian coordinates and polar coordinates. So what we can do is simply analyze our equation. It says R equals negative to cosine theta. Plus 6 sine theta. If we multiply both sides by r, we're going to get r squared equals. Negative to our cosine theta. Plus 6 are sin theta. And this is really useful because now we have our cosine theta, our sine theta. And we also know that R2d can be written as X2 + Y2. This is an additional formula that we should know. In polar coordinates. So now we can replace R squared with X2 + Y squad on the left hand side. On the right-hand side, we have -2, and our cosine theta is basically X. Plus 6, our sin theta is Y. So, we get an equation in a form of X2 + Y2 equals -2 X + 6 Y. What we can do is simply move all of our terms to the left. Which gives us X2 + 2 X. Plus y2 minus 6 Y equals 0. And then we can complete the square for each part starting with X we get. X + 1 squad, right? Because we have a coefficient of 2 in front of X, we have to divide by 2 to get 1. So now if we square, we're going to have X2 + 2 X + 1 squad. So what we want to do is simply subtract 1. To return the original expression X2 + 2 X, right? And now considering a Y2 minus 6 Y. We're going to add Y minus 6 divided by 2, which is 32, and we want to subtract 3 squad, so we're subtracting 9. This is equal to 0. So now, combining our constants, we get negative 10, which we can move to the right. And we can then rewrite the result on the equation as X + 1 squad. Plus y minus +32 equals sum. So this is our equation. We can label it. It is part of our final answer, and we are going to describe the curve. Notice that it has a form of x minus X02 plus Y minus Y +02 equals R2, which is an equation of a circle with a center. X0 Y 0. So we can say that this is a circle. With a center. At Now, notice that it says X + 1, which we can rewrite as X minus -1. So the coordinates of the center would be -1.com. We So we have a circle, it's center, and the radius would be. On the right hand side, let's remember that 10 is equal to R2, right? It is the square of the radius. So the actual radius would be. Square root of sun. And that would be our final answer for this problem. Thank you for watching.

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.

r = 6 cos θ + 8 sin θ

Key Concepts

Polar and Cartesian Coordinate Systems

Conversion Formulas Between Polar and Cartesian Coordinates

Identifying and Describing Curves from Equations

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