Given Information:
Focus of the ellipse: (3, 4)
Directrix equation: X - Y - 1 = 0
Eccentricity: 2

Understanding:
To find the coordinates of the extremities of the major axis of the ellipse, we need to understand the properties of an ellipse and how it is related to its focus, directrix, and eccentricity.

Properties of an Ellipse:
1. An ellipse is a set of points in a plane, the sum of whose distances from two fixed points (the foci) is constant.
2. The line passing through the foci is called the major axis of the ellipse.
3. The distance between the foci is 2a, where 'a' is the semi-major axis.
4. The eccentricity of an ellipse is given by the ratio c/a, where 'c' is the distance between the center and either focus.

Solution:
To find the coordinates of the extremities of the major axis, we need to determine the center, semi-major axis, and eccentricity of the ellipse.

1. Finding the Center:
Since the focus of the ellipse is given as (3, 4), the center of the ellipse is also at (3, 4).

2. Finding the Semi-Major Axis:
The semi-major axis 'a' can be found using the eccentricity 'e' and the distance 'c' between the center and focus.
Given that the eccentricity 'e' is 2, we can use the formula e = c/a to find 'c'.
Since 'c' is the distance between the center and either focus, we can calculate it using the distance formula:
c = √[(x2 - x1)² + (y2 - y1)²]
= √[(3 - 3)² + (4 - 4)²]
= √[0 + 0]
= 0

Now, we can substitute the values of 'e' and 'c' into the formula e = c/a to solve for 'a':
2 = 0/a
a = 0

3. Conclusion:
The center of the ellipse is (3, 4), and the semi-major axis 'a' is 0. This means that the ellipse is degenerate and appears as a single point at the center (3, 4).

Explanation:
Since the semi-major axis 'a' is 0, the ellipse degenerates into a single point at the center. The major axis, which is the line passing through the foci, also disappears. Therefore, there are no extremities of the major axis in this case.

Summary:
The coordinates of the extremities of the major axis cannot be determined because the semi-major axis 'a' is 0, resulting in a degenerate ellipse.

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