To find the general equation of the ellipse, we can start by considering the standard form of an ellipse centered at the origin:
(x^2 / a^2) + (y^2 / b^2) = 1,
where 'a' represents the semi-major axis and 'b' represents the semi-minor axis of the ellipse.
In this case, the center of the ellipse is given as (1, 2), so we need to shift the origin accordingly. We can achieve this by subtracting the x-coordinate of the center (1) from the x-coordinate of any point on the ellipse, and subtracting the y-coordinate of the center (2) from the y-coordinate of any point on the ellipse.
Now, let's find the values of 'a' and 'b' based on the given information:
The distance between the center (1, 2) and the focus (3, 2) is equal to 'c', which represents the distance from the center to the foci. In this case, 'c' is 2 units (since the x-coordinate of the focus is 2 units greater than the x-coordinate of the center).
The distance between the center (1, 2) and the vertex (5, 2) is equal to 'a', which represents the semi-major axis. In this case, 'a' is 4 units (since the x-coordinate of the vertex is 4 units greater than the x-coordinate of the center).
The semi-minor axis 'b' can be found using the relation: b^2 = a^2 - c^2. Substituting the values, we have b^2 = 4^2 - 2^2 = 16 - 4 = 12. Taking the square root, we get b = sqrt(12) = 2 * sqrt(3).
Now we can write the equation of the ellipse in general form, considering the shifted center:
((x - 1)^2 / 4^2) + ((y - 2)^2 / (2 * sqrt(3))^2) = 1.
Simplifying further, we have:
(x - 1)^2 / 16 + (y - 2)^2 / 12 = 1.
Therefore, the general equation of the ellipse centered at (1, 2), with a focus at (3, 2), and a vertex at (5, 2), is ((x - 1)^2 / 16) + ((y - 2)^2 / 12) = 1.