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. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)

Accepted Answer

Hello, everyone in this video, we're gonna be looking at this practice problem where we want to find the equation of an ellipse with the conditions that it has a major axis, horizontal length of 22 a length of the minor axis of 18. And it is centered around the origin or 00. And in order to find the equation for the ellipse, let's remember the standard form for the equation of ellipses. So ellipses follow the form or the equation for ellipses, follow the form H or X minus H squared divided by A squared plus Y minus K squared divided by B squared is equal to one. In this case, our ellipse is centered around the origin and remember that the center of an ellipse is the H and K value. So really, we can simplify our standard equation to just be X squared divided by A squared plus Y squared, divided by B squared is equal to one. And if I were to draw and ellipse or our elis were given that the horizontal length, this distance here is 22 and the length of the minor axis is 18. So I'll draw this in blue. And we're gonna say that this is the center at 00. So in order to solve for our standard form, we need the values of A and B and recall that A and B are actually the values of the center to the outermost points along the major and minor axis. Where for B A, we're looking from center to outer point on the major axis. And for B, we're looking for center to outer point on the minor axis. And we're actually given the length of all the minor and major axis as I've drawn them in this ellipse. So I made the ellipse a little bit bigger. So it's easier to see. So E would be along the major axis here and it's from center to the outer point. So this would be A but this going to the left would also be A. So you can see that the total length of the major axis is actually two A. So I can say two A is actually equal to 22 because the total length of the major axis gives me two A's. And from here, I can solve for the value of A by dividing by two. So that gives me A is equal to 11. Now, if we look at the minor axis, you can see that I have A B from center to this outermost point going up and another bee from the center going down. So once again, I have two B is equal to the total length of the minor axis and the minor axis is 18. And I can solve for B by dividing by two. So that leaves me with B is equal to nine. Now, I can plug both of these values into my Standard equation. So I have X squared divided by A squared. So A is equal to 11, so 11 squared plus Y squared divided by B squared which is nine. So we'll have Y squared divided by nine squared, nine squared. S equal to one. If I simplify that I have X squared divided by 11 times 11 is 1 21 plus Y squared divided by nine times nine, which is 81 that's equal to one. So this is the standard form equation for my lips. And you can see that this aligns with answer choice C so hopefully this video helped and see you next time.

In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Major axis horizontal with length 8; length of minor axis = 4; center: (0, 0)

Key Concepts

Ellipse Standard Form

Major and Minor Axes

Center of the Ellipse

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