Final answer:
The equation of the ellipse with the given vertices at (4, 3) and (4, 9) and focus at (4, 8) is (x-4)²/8 + (y-6)²/9 = 1.
Step-by-step explanation:
The student is asking for the equation of an ellipse with given vertices and a focus. The vertices provided, (4, 3) and (4, 9), indicate that the major axis is vertical. The distance between these vertices is the length of the major axis, which is 6 units, hence the semi-major axis a is 3 units. The focus point (4, 8) is 1 unit away from the center of the ellipse, which is located at (4, 6), so the focal length c is 1 unit. Using the relation c² = a² - b², where b denotes the length of the semi-minor axis, we can find b. Knowing that c = 1 and a = 3, we get b² = 3² - 1² = 9 - 1 = 8, hence b = √8. The standard form equation of an ellipse with a vertical major axis is ∑(x-h)²/b² + (y-k)²/a² = 1, where (h, k) is the center of the ellipse. Therefore, the equation of the ellipse is (x-4)²/8 + (y-6)²/9 = 1.