Answer :
Final answer:
The equation of the ellipse is A) 25x² + 169y² = 4225.
Explanation:
To find the equation of an ellipse, we need to know the center, vertices, and one focus of the ellipse. Since the center is given as (0,0), we can use the distance formula to find the distance between the center and the vertex:
d = √((x2 - x1)² + (y2 - y1)²)
Plugging in the values, we get:
d = √((0 - 13)² + (0 - 0)²) = √169 = 13
Therefore, the major axis of the ellipse is 2a = 26.
The equation of the ellipse with center (0,0) and major axis 26 is:
x²/a² + y²/b² = 1
Since the focus is (12,0), we know that a = 13, so we can substitute the values into the equation to get:
x²/13² + y²/b² = 1
Simplifying the equation, we get:
x²/169 + y²/b² = 1
Since the vertex is (13,0), we know that the distance from the center to the vertex is a, which is 13. The value of b can be found using the equation b² = a² - c², where c is the distance from the center to the focus:
b² = a² - c²
b² = 169 - 12²
b² = 169 - 144 = 25
So, the equation of the ellipse is:
25x² + 169y² = 4225
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