# 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.

Finding the Vertices of an Ellipse Given Its Foci and a Point on the Ellipse
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.
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spornya1
Step 1
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
${b}^{2}/a=7,\phantom{\rule{thinmathspace}{0ex}}{a}^{2}-{b}^{2}=144\to {a}^{2}-7a-144=\left(a-16\right)\left(a+9\right)=0$
Step 2
The ends of the required ellipse are at $\left(±16,0\right)$ on x-axis
The ends of the (not asked for) hyperbola are at $\left(±9,0\right).$