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. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)

Accepted Answer

Hello today we're going to be finding the equation of an ellipse when it's not centered at the origin. So the question states that we have an ellipse with the major axis that is vertical with the length 10 and the length of the minor axis is six. With the center being at two comma negative four. We need to find the standard equation of this ellipse. Now one thing to note is that this center is not at the origin. So we need to find the equation of an ellipse that's not at the origin. And there's two possible forms of that equation. So if we have the equation of the lips where the x axis is the transverse axis, then the form of that ellipse is going to be x minus H squared over a squared plus y minus k squared over B squared is equal to one. And if our y axis ends up being our transverse axis then the equation is going to be very similar except a squared and b squared are going to be swapped. So we have x minus H squared over B squared plus y minus k squared over a squared equal to one. So we need to figure out which form of the ellipse we're looking for. So let's just quickly recall what the information is given. We are told that the vertical major axis has a length of 10. So if we draw this on axis, the center is gonna be a two comma negative four. So roughly here and that means that the vertical length is going to be greater than the minor axis and vertical means up and down. So the lips is gonna have a rough shape just like this where the major axis is respect to the y axis. So since the major axis is with respect to the y axis, the form of the equation we're looking for is going to be the second option right here. Now we are told that this vertical length has a length of 10, so the total vertical length is going to be 10 and the length of the minor axis which in this case is gonna be respect the X axis. That length is going to be six. Okay, so there's four things we need to find, we need to find our H r k r b squared and R a squared. Now keep in mind that a squared and B squared represent the distance from the center to one of the verses of the lips. So a squared is going to be our major axis distance. So A is only going to be the distance from the center of the ellipse to the major versus E. Which is going to be half the distance of the total length. So A is going to equal to five. And the same logic applies to B. B represents the distance from the center of the ellipse to the minor verte asi. So that's gonna be half the distance of the minor axis. So B is going to be equal to three. Now we want to go ahead and get a squared and B squared. So squaring both sides of this equation is going to give me a squared equal to five squared which is 25 B squared equal to three squared which is nine. So now we have our values for A. And B. But now we need to get our values for H and K. Well since the center is not the origin, the center is of the form H comma K. So the center is given to us as two comma negative four, so H is equal to two and K is equal to negative four. Now we have the four variables. We need to construct the equation. So let's go ahead and do that. So putting these together is going to give me x minus H which is to that quantity squared over B squared which is nine plus y minus K, which is negative four. And keep in mind it's minus negative four. So we're actually gonna have y plus four squared Over a squared, which is 25 equal to one. And that's gonna be the standard form of the equation of the Ellipse. That also means that the answer to this problem is going to be be So I hope this video helps you in understanding how to find the standard form of an equation of a lips and I'll go ahead and see you all in the next video

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