Final answer:
To find the coordinates of the center of the circle, you can rearrange the equation of the line in the standard form of a line equation, substitute it into the equation of the circle, and solve for the unknown variable. In this case, the center of the circle is located at (3, 8.125).
Step-by-step explanation:
To find the coordinates of the center of the circle, we can start by rearranging the equation of the line in the standard form of a line equation, which is y = mx + c.
In this case, the equation 2x + y - 5 = 0 can be rearranged to y = -2x + 5.
Next, we substitute this equation into the equation of the circle to find the coordinates of the center.
(x-3)² (y-p)² = 5
Replacing y with -2x + 5, we get:
(x-3)² (-2x + 5 - p)² = 5
Since the line is a tangent to the circle, the discriminant of this equation should be equal to 0.
By expanding and simplifying, we have:
4x² - 20x + 25 - 4px + 6x² - 30x + 25 = 5
Combining like terms, we get:
10x² - 50x + 45 - 4px = 0
For the discriminant to be 0, we have:
(-50)² - 4 * 10 * (45 - 4p) = 0
Simplifying further, we obtain:
2500 - 40(45 - 4p) = 0
Expanding and simplifying, we have:
2500 - 1800 + 160p = 0
Combining like terms, we get:
160p = 1300
Solving for p, we find:
p = 1300/160 = 8.125
Therefore, the center of the circle has coordinates (3, 8.125).