To solve this problem and find if the statement is true, we can start by remembering the general equation for an ellipse:
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}=1[/tex]
where a and b are the major axis and the minor axis for the ellipse, as shown in the following diagram:
The foci of an ellipse, are the focal points of the ellipse, for reference, we show them in the diagram:
The foci can be found using the following equation:
[tex]f=\sqrt[]{a^2-b^2}[/tex]
To prove if the foci f is really 1/3 of the length of the major axis (in this case the major axis is a) we can give random values to a and b.
Values for a and b:
a=6
b=4
If the foci of the ellipse were located at 1/3 of the length of the major axis, we should find that the foci "f" is 1/3 of 6, thus, we should find that f=2, let's see if that is true by substituting a and b into the formula for f:
[tex]f=\sqrt[]{a^2-b^2}[/tex][tex]f=\sqrt[]{6^2-4^2^{}}[/tex][tex]\begin{gathered} f=\sqrt[]{36-16} \\ f=\sqrt[]{20} \end{gathered}[/tex]
solving the square root we find the value of f:
[tex]f=4.47[/tex]
Instead of 2 (which would have been 1/3 of the major axis), we find that f is 4.47, thus the statement "The foci of an ellipse are located at 1/3 the length of the major axis from both sides of the center." Is NOT TRUE.
Answer: False
