The question is as follows:
The focal points of an ellipse are $(12,0)$ and $(−12,0)$, and the point $(12,7)$ is on the ellipse. Find the points where this curve intersects the coordinate axes.
I know that the center of the ellipse would be $(0,0$) because that is the midpoint of the foci. However, I am not sure as to how this information will help me in finding the intersections on the coordinate axes (or the vertices). Any help will be greatly appreciated.
The sum of the distances from a point on the ellipse to its foci is constant. You have both foci and a point, so you can find the sum of the distances. Then you can find the vertices since they are points on the ellipse on the $x$-axis whose sum of distances to the foci are that value.
The $7$ in $y$ coordinate of $(12,7)$ is $7$, the length of semi-latus rectum; Also $c$ is $12$ and $c^2= 144 = a^2-b^2$ So we have two equations
$$ b^2/a= 7,\, a^2-b^2 =144 \rightarrow a^2- 7a - 144 = (a-16)( a+9)=0 $$
The ends of the required ellipse are at $ (\pm 16,0) $ on x-axis
The ends of the (not asked for) hyperbola are at $ (\pm 9,0). $
