Question

What is the equation of the ellipse with foci (5, 0), (-5, 0) and co-vertices (0, 4), (0, -4)?

Answer 1

The elipse is the geometric body composed by all the points such that the sum of their distance to two fixed points called "focis" is constant.
If you apply this definition to a ortonormated referential system, that is a normal XoY system, and we put the focis on any axis, this time being the normal x-axis, we will optain the equation:
\[\frac{ x^2 }{ a^2 }+\frac{ y^2 }{b^2 }=1\]
Where \(a,b\) are the semi axis of the elipse.
Now, there is something fishy about this and I can imagine you might understand, that the equation of the elipse is not often straight forward to obtain, in contra-example to the line and the circle, we need to find the mayor and minor axis of this equation in order to plot it and thereby solve this.

Now, we got the minor one, the co-axises often determine the minor axis, from which we can conclude that \(b= 4\), and in order to help us find the mayor axis "a", we will help us with the equation:
\[a^2=b^2+c^2\]
Where "c" is the focal distance, and "a" being the mayor axis we are looking for.
plotting the information in:
\[a^2=(4)^2+(5)^2\]
\[a=\sqrt{41}\]
Now, we have found the mayor axis we can plug them in the canonic equation of the elipse:
\[\frac{ x^2 }{ a^2 }+\frac{ y^2 }{ b^2 }=1\]
\[\frac{ x^2 }{ (\sqrt{41})^2 }+\frac{ y^2 }{ (4)^2 }=1\]
\[\frac{ x^2 }{ 41 }+\frac{ y^2 }{ 16 }=1\]

Answer 2

I see, I think. o: Thank you very much! ♥

Answer 3

@Owlcoffee what about this one?



What is the equation of the ellipse with vertices (0, 10), (0, -10) and co-vertices (3, 0), (-3, 0)?

Answer 4

The process is pretty much the same, the vertices determine the mayor axis and the co-vertices determine the minor axis.
therefore "a" and "b" are already given ,so you just have to plot.

Answer 5

so a = 10 and b = 0?

Answer 6

Correct.