Answer:
(x -3)²/9 +(y -2)²/5 = 1
Explanation:
You want the equation of the ellipse centered at (3, 2) with foci at (1, 2) and (5, 2), and a = 3c (where a is the major axis).
Ellipse equations
The relevant equations are ...
(x -h)/a² +(y -k)/b² = 1 . . . . . . a, b are the semi-axes, (h, k) is the center
b² +c² = a² . . . . . . . . . . . c is the distance from the center to a focus
The distance from the center (3, 2) to a focus (5, 2) is c = 5-3 = 2 units. The problem statement tells us that 2a = 3c = 3·2 = 6, the length of the major axis.
Then the value of b² is ...
b² = a² -c² = (6/2)² -2² = 5
The ellipse centered at (3, 2) with semi-major axis 3 and semi-minor axis √5 will have equation ...
(x -3)²/9 +(y -2)²/5 = 1
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Additional comment
In the problem statement, 'a' is defined as the length of the major axis. In our answer, we have used 'a' as half that length. That is why our equation for c is 2a=3c, rather than the a=3c relation given in the problem statement. This makes sense if you consider the different definitions for 'a'.