By definition, a+b always equals the major axis length QP, no matter where R is. So a+b equals OP+OQ.
So b must equal OP. (And a equals OQ).
An ellipse has two focus points. The word foci (pronounced 'foe-sigh') is the plural of 'focus'. One focus, two foci.
The foci always lie on the major (longest) axis, spaced equally each side of the center. If the major axis and minor axis are the same length, the figure is a circle and both foci are at the center. Reshape the ellipse above and try to create this situation.
Note how the major axis is always the longest one, so if you make the ellipse narrow, it will be the vertical axis instead of the horizontal one.
In the figure above, drag any of the four orange dots. This will change the length of the major and minor axes. You will see how the foci move and the calculation will change to reflect their new location.
Given an ellipse with known height and width (major and minor semi-axes) , you can find the two foci using a compass and straightedge.
The underlying idea in the construction is shown below. The point R is the end of the minor axis, and so is directly above the center point O,
and so a = b.
By definition, a+b always equals the major axis length QP, no matter where R is. So a+b equals OP+OQ.
So b must equal OP. (And a equals OQ).
The construction works by setting the compass width to OP and then marking an arc from R across the major axis twice, creating F1 and F2.. For more, see
If the inside of an ellipse is a mirror, a light ray leaving one focus will always pass through the other. For more on this see Optical Properties of Elliptical Mirrors