Final answer:
The center of the circle is (3, -1) and the radius is 5. The equation of the tangent line at (7, 2) is 4x + 3y - 34 = 0.
Step-by-step explanation:
To find the center and radius of the circle, we can rewrite the equation of the circle in the form (x - h)² + (y - k)² = r². By completing the square, we get (x - 3)² + (y + 1)² = 25. This means the center of the circle is at (3, -1) and the radius is √25 = 5.
To find the equation of the tangent at (7, 2), we can find the slope of the tangent line and use the point-slope form of a line. The slope of the tangent line is -1 divided by the slope of the radius from the center of the circle to the point of tangency. The slope of the radius is (2 - (-1)) / (7 - 3) = 3/4. Therefore, the slope of the tangent line is -4/3. Using the point-slope form, we have y - 2 = (-4/3)(x - 7), which simplifies to 3y + 4x = 34. Rearranging the equation, we get 4x + 3y - 34 = 0.