Find the standard form of the equation of the ellipse satisfying ...

Question

Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (-4,0), (4,0); Vertices: (-5,0) (5,0)

Accepted Answer

Hello today we're going to find the standard form of the equation of an ellipse. So we are told that this ellipse has the vertex is and negative eight comma zero and eight comma zero. And it's fosse are located at negative three comma zero and three comma zero. So before we jump into this let's just go ahead and graph are given information. So we are told that the vert is's are negative a comma zero and a comma zero. So let's just go ahead and draw that real quick. A comma zero And -80. We are also told that the focus I. R. At negative three comma zero and three comma zero. So we can go ahead and draw that real quick. And we can also represent the values of the fosse with the variable C. So since our foes I lie on the X axis that means that the major axis or the transverse axis is going to be the X axis. And that means that the equation of this ellipse is gonna have the form X squared over a squared plus Y squared over B squared is equal to one. So that means we could that means a positive or -10 is going to be our major versus is. So we can represent that with variable a. And our lips is roughly gonna have this shape. So what we need to do is go ahead and figure out what the values of our minor access are going to be. So let's go ahead and do that. Well we have an equation for an ellipse that allows us to equate C A and B together and that's going to be C squared is equal to a squared minus B squared, but we don't have C squared A squared, but we can figure those out because we have values for C and a C is just the value of the fosse, which is going to be three and A is the value of the major vertex is which is going to be eight squaring both sides of the sea equation is going to give me c squared equal to three squared which is nine. And squaring both sides of the equation, is going to give me a squared is equal to eight squared which is 64. So plugging this in to our equation here is going to give us nine equal to 64 -7 square. Now, in order to solve for b squared, I'm gonna go ahead and subtract 64 from both sides of this equation. Doing so is going to give me negative 55 equal to negative b squared. And in order to solve for B squared, I'm gonna go ahead and divide both sides by negative one. And that gives me B squared equal to 55. So now we have the two values that we need to construct the equation. So let's go ahead and plug that in now. So the equation of the ellipse is going to be X squared over A squared, which is 64 plus Y squared over B squared, which is 55 equal to one. And that's going to be the standard form of the equation of this ellipse. And that tells us that the answer to this problem is going to be a So I hope this video helps you in understanding how to find the equation of an ellipse, and I'll go ahead and see you all in the next video.

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