Ellipse graph from standard equation | Precalculus | High School Math | Khan Academy

Whereas which ellipse is represented by the equation ( (x - 4)^2 / 16 + (y - 1)^2 / 49 = 1 )?

And we're given a bunch of choices here. We're given four choices here, so let's just think about what's going on here.

The center of the ellipse is going to be ( 4, 1 ). How do I know that? Well, the equation of the ellipse is going to be ( (x - \text{(x coordinate for the center)})^2 / \text{(horizontal radius)}^2 + (y - \text{(y coordinate of the center)})^2 / \text{(vertical radius)}^2 ).

So the center is going to be ( 4, 1 ). The center here is not ( 4, 1 ). The center over here is not ( 4, 1 ). Not ( 4, 1 ). The only choice that has a center at ( 4, 1 ) is this one over here.

So we already know this. This is the choice without even looking at the horizontal and the vertical radius. But we can verify that this works out because a horizontal radius right over here—notice it goes, this orange line which can represent the horizontal radius—it has a length of 4.

And so the horizontal radius is 4, and so we see indeed that 16 is the horizontal radius squared. This is ( 4^2 ). And if we look at the vertical radius here, we see it has a length of 7.

We're going from ( y = 1 ) to ( y = 8 ); it has a length of 7, and we see in that equation that this indeed is ( 7^2 ). So that was pretty straightforward.