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.

Trace Glass 2022-10-21 Answered
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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Answers (1)

spornya1
Answered 2022-10-22 Author has 18 answers
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 , a 2 b 2 = 144 a 2 7 a 144 = ( a 16 ) ( a + 9 ) = 0
Step 2
The ends of the required ellipse are at ( ± 16 , 0 ) on x-axis
The ends of the (not asked for) hyperbola are at ( ± 9 , 0 ) .
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