In Exercises 45–52, use your answers from Exercises 41–44 and ...

Question

In Exercises 45–52, use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line.

Ellipse: Center: (−2,3); Vertices: 5 units to the left and right of the center; Endpoints of Minor Axis: 2 units above and below the center

Accepted Answer

Welcome back everyone. In this problem, we want to determine the parametric equations for the ellipse centered at 05 with vertices four units opened down from the center and end points of the major axis seven units to the left and right of the center. A says X of T equals four CT and Y of T equals five plus seven sine TB says there are seven CT and five plus four sine T respectively, C five plus four CT and seven sine T and D five plus seven CT and four sine T. Now let's just try to sketch our ellipse here just to make sure we understand exactly what we are looking for. OK. So let's say this is our ellipse and we know that at the center of our ellipse, the coordinate is 05. OK? Where its vertices are four units up and down from the center. So this is four units in both directions and left and right of the center. It's seven units. OK. So we have four units and seven units. Now, what do we know that can help us to figure out the parametric equations for the ellipse? Well, recall, OK? Not for an ellipse. OK. For an ellipse centered at, oops, let me, let me spell that properly here centered at HK OK. With a semi major axis, semi major axis. I don't know, I'm not spelling so well today, but semi major axis of our units. OK. Of our units. Then we know that the, the equation for X of T, OK. X of T is going to be equal to age plus RRKT. OK. And let's put that in a bracket here, Y of T, OK is going to be equal to K plus R sine of T. OK. So to find the parametric equations, we just need to think about our respective coordinates and the respective semi major axis. Let's start with X. OK. Now, for our problem here in our problem, we are told that it's centered at +05. OK. So we can say that H is zero and K is five, but R is going to depend on what axis we're looking at. So for X, OK, if we think about X here, OK, then the end points of the major axis for X axis are seven units to the left and the right of the center. OK. So that means the major axis length is 14 units giving a semi major axis length of seven. So X of T, OK. X of T is going to be equal to H which is zero plus the value here for a semi major axis seven K multiplied by the cosine of T, which works out to seven CA T. On the other hand, for Y of T, we know that K is equal to five but no, it's semi major axis is four semi. For here, for the semi minor axis, it's four because these vertices are four units up and down from the center. So the minor axis length is eight units giving a semi minor axis length of four. So that's gonna be five plus four sine of T. Therefore, X of T is seven KT and Y of T is five plus four sine T. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Parametric Equations of Conic Sections

Ellipse Geometry and Parameters

Using Center and Axis Lengths to Form Parametric Equations

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