In Exercises 25–36, find the standard form of the equation of eac...

Question

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to find the equation of an ellipse with the properties of having folk I at negative 50 and 50. And why intercepts of negative seven and positive seven. So the first thing I'm gonna do is try to visualize this ellipse by drawing it. So I'm going to draw the Y axis and then the X axis. And The First Book I is at -50. So I'll draw a dot there and say it's negative five zero. My second book, I is that positive 50. So I'll draw another dot there and the Y intercepts are negative seven. So the coordinate for that would be zero negative seven and zero positive seven. So I can sort of draw an ellipse passing through all of these points. Now I sort of have a visual for my lips but I can't write the equation if I don't know the form to follow. So let's see if this, let's find the center of this ellipse to see if I can follow the standard form for an ellipse centered at the origin. In order to find the value of the center. I can use the folk. I because the folk, I are actually distanced equal distance from the center and actually this wouldn't be my ellipse. It would go a little bit past the focus just like that. And the folk I r distance equally from the center. So if I wanted to find the center, I need to find the midpoint between their distance. So for the first book I I start at negative five And the 2nd 4 guy ends up five. So add five and divided by two because I want the midpoint. So now I have negative five plus five divided by two. So it becomes zero divided by two or just zero. So now I know that my center is or has a X. Value of zero. And because my first folk I my first focus and my second focus have the same Y value. I know that my center will also have that same Y. Value of zero. And I know that it will have the same Y value because my center follows the form H. K. And my okay follow the form H plus or minus C. Depending on which way they're going. Okay so honestly both have K. As the why value. And in this case that y value is zero. So now that I know that my lips is centered at the origin, I can use the simplified version for the standard form of an ellipse where the standard form is X squared divided by a squared plus Y squared divided by b squared Is equal to one. And note here I need to find the value of A. And the value of B. So recall that A. Is the total distance along the major axis. And this case the major axis is the horizontal axis because that is the line along the two. So this red line is the total distance and a. Is the value from the center to the outer point. So for a I'm looking at going from the center to the outer point along the major axis. So notice how the total length of the major axis actually gives me to A. And B is the same thing going from the center to the outer point except along the minor axis. And in this case the minor axis is the vertical value. So using that I can determine that the value of V is actually seven. Because going from the center to this outer point, I go from 0-7. So I know that B is equal to seven. And I have no way of determining this outer point for A. Except that I can use the formula C squared is equal to a squared minus B squared. And we haven't defined the value of C. But we were given it through the folk. I so if I have a focus, My first focus is negative five zero. And that is a church minus C. K. I know that K. Is equal to zero and h minus C is equal to negative five. Since h is equal to zero because my center is at the origin or 00. This becomes negative C. Is equal to negative five Or C. is equal to five. And now I have both B and C. To find I can use that to find the value of A. So if I sell for A here I'll have A squared is equal to C squared plus B squared. I'll take the square root of both sides. So now I have A. Is equal to the square root of C squared plus B squared. Now I can plug in so I have A. Is equal to the square root of C squared which is five squared plus B squared which is seven squared. And if I solve that I have a. Is equal to the square root of 25 plus and 25 plus 49 is equal to 74. So I have a. Is equal to the square root of 74. And now I finally have a value for B. And for A. So I can plug that into my standard form. So now I have X squared Divided by the square root of 74 squared plus Y squared divided by seven squared is equal to one. And if I simplify that I have X squared divided by 74 because the square root of 74 times the square root of 74 is just plus. Why squared Divided by 49? since seven times 7 is 49 Is equal to one. So this becomes my final equation from my lips. And you can see that this aligns with answer choice D. So hopefully this video helped and see you next time

Struggling with College Algebra?

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-2, 0), (2, 0); y-intercepts: -3 and 3

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