Graph the ellipse and locate the foci.

Question

\frac{x^{2}}{36} + \frac{y^{2}}{25} = 1

Accepted Answer

Hello. Today we're gonna be using the given equation to find the graph of the ellipse. So what we are given is x squared over 49 plus Y squared over 16 is equal to one. Now this graph is given to us in standard form but notice that the denominator of the X term is greater than the denominator of the white term. That means that the standard form of this equation is going to be x squared over a squared plus y squared over B squared is equal to one. And this is where the major axis of the ellipse is going to be the X axis. So we're going to have a horizontal ellipse furthermore, this is the standard form of the equation of ellipse where the center is located at the origin. So the center is going to be at zero comma zero. So now that we have this information, we need to identify the vertex is for the major and minor axis for the major axis. The vertex is are denoted by the A variable And here a squared is equal to 49. If we take the square root of both sides of this equation, we get a is equal to plus or minus seven. Now keep in mind that the major axis is located at the X axis. So that means that the vertex is are going to be located at seven comma zero and negative seven comma zero. Now let's go ahead and identify the vertex is for the minor axis, which is denoted by the b variable. Here B squared is equal to 16. And taking the square root of both sides is going to give us B is equal to plus or -4. So since the minor axis is located at plus or minus four and the minor axis is the y axis, the location of the vertex is are going to be at zero comma four and zero comma negative four. Now we need to identify the fosse. The equation of the fosse is given to us by C is equal to the square root of a squared minus B squared. And we already know the values of a squared and B squared. It's 49 16. So if we plug in those values into this equation we get C is equal to the square root of 49 minus 16, which simplifies to the square root of 31. Now the square root of 31 cannot be simplified any further. So we're going to leave it as plus or minus the square root of 31. Now the focus I lie on the major axis of the ellipse, which in this case is the X axis. So that means that the locations of the fosse are going to be at the square root of 31 comma zero and negative square root 31 comma zero. So now that we've identified all the information, we need for the ellipse. We need to select the graph that accurately represents all this information and the graph is going to be option C, The reason why is because we have an ellipse whose origin or starting point is at zero comma zero. The vert is's are at seven comma zero and negative seven comma zero for the major axis. For the minor axis we have the vertex is at zero comma four and zero comma negative four. And we have the faux sign which are going to be roughly located here at the square root of 31 comma zero And - 00. So with that being said, the answer to this problem is going to be C. So I hope this video helps you in understanding how to create the graph of an ellipse and I'll go ahead and see you all in the next video.

Graph the ellipse and locate the foci. $\frac{x^{2}}{36} + \frac{y^{2}}{25} = 1$

Key Concepts

Standard Form of an Ellipse

Graphing an Ellipse

Locating the Foci of an Ellipse

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