Question

Help please????
Write the equation of an ellipse with vertices (0, 5) and (0, -5) and co-vertices (2, 0) and (-2, 0).
Thank you!!! <3

Answer 1

Okay, since we have already the major and minor axis, we can express the equation of any elipse with equation:
\[\frac{ x^2 }{ a^2 }+\frac{ y^2 }{ b^2 }=1\]
Where "a" is the major axis and "b" the minor axis.
We can tell by the points given that the major axis is parallel to the y-axis and the minor axis is parallel to the x-axis.
Now, we need the distance of these two axis, and we can tell by the very coordinates of the points of the vertices, because they have to be symmetrical (most of the cases) so therefore, we can tell that a=5 and b=2.
So, let's plug it in the equation:
\[\frac{ x^2 }{ (5^2) }+\frac{ y^2 }{ (2^2) }=1\]
And simplifying:
\[\frac{ x^2 }{ 25 }+\frac{ y^2 }{ 4 }=1\]
As you can observe, we don't need to know the center of the elipse, the focal distane or the excentricity in order to find the equation of any elipse, all we need is the major and the minor axis, which will never be so obvious in any excercise.
In this case, the center is (0,0) and the equation I used is called the "canonic" form of the elipse.