Graph each plane curve defined by the parametric equations for...

Question

Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.

x = 4 sin t , y = 3 cos t

Accepted Answer

Welcome back everyone to another video. Given the parameter equations X equals 5 cosine T and Y equals 2 T for T in the interval between 0 and 2 pi inclusive, grab the corresponding plane curve. What is the rectangular equation of this curve? So we are given the X of 5 cosine T, right? And we know that Y is equal to 2 sin T. What we're going to do is solve for sine and cosine. The reason why we're doing this is because sine squared T plus cosine squaret based on Pythagorean identity is equal to 1. So if we solve for sine and cosine, we can square those substitute into the equation, and we're going to get 1, then this will allow us to eliminate the parameter. Now, from the first equation, we have to divide both sides by 5 and this gives us cosine. For the second equation, we divide both sides by 2, and this gives us sign. When we square them, we're going to get X divided by 5 squared, plus Y divided by 2. Squared must be equal to 1 and now if we distribute the squares, we're going to get x 2 divided by 25 plus y divided by 4 we're going to square the Y and this is going to give us 1, and this is the general equation of an ellipse, right? X2d divided by a quad. Plus y2 divided by b2 is equal to 1 and of course we can see that A is equal to 5 if we take a positive number and B is equal to 2. Now the center of this ellipse is going to be at a 0.00. That's because we're not subtracting or adding any numbers to X or Y, right? So we're going to have a center at the origin. And now looking at the value of A, which corresponds to our major axis, right? Essentially it tells us that our ellipse will have a major axis with a length of 2A, which is 10, right, because essentially we have a of 5, the length of the major axis is 10. Let's call it major. And for the minor axis because B is 2, well, essentially 2 multiplied by 2 gives us 4. So the length for the minor axis is 4. So what we want to do is basically illustrate. Our XY plane. We want to make sure that the center of the ellipse is at the origin looking at the major axis, it's length is 1, right? So if we have a length of 10, we are going to divide it by 2. We're going to get the 5, so we have to add 5 units to the right, 5 units to the left. And then for the minor axis, the length is 4, right, with the center at 0, so this gives us. Intercepts at 2 and 2. As we can see, the overall length is 4, and the overall length for the major axis is 10. And now we simply want to connect those points into the shape of an ellipse, right? We essentially want to make a rough sketch and now let's illustrate. The complete graph using a graphing utility. So we have our ellipse with the center at 00, and we can see the intercepts, right? -55, and then -2 and 2, and that's our final answer. Thank you for watching.

Graph each plane curve defined by the parametric equations for t in [0, 2π] Then find a rectangular equation for the plane curve. See Example 3.

x = 4 sin t , y = 3 cos t

Key Concepts

Parametric Equations

Graphing Techniques

Rectangular Equation

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