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.
On
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). $
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.