Final answer:
The value of p in the circle's equation x² + y² - 4x + 6y + p = 0, given that the radius is 3, is found by completing the square and comparing it to the standard form of a circle's equation. By doing this, we find that p = 4.
Step-by-step explanation:
To find the value of p in the given equation of the circle x² + y² - 4x + 6y + p = 0, we first need to complete the square for both x and y terms:
- Group the x terms and y terms: (x² - 4x) + (y² + 6y) = -p.
- Complete the square for the x terms:
(x - 2)² = x² - 4x + 4. - Complete the square for the y terms:
(y + 3)² = y² + 6y + 9. - Substitute these back into the equation and balance it:
(x - 2)² + (y + 3)² = 4 + 9 - p. - Rewrite the equation:
(x - 2)² + (y + 3)² = 13 - p.
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius. Given that the radius is 3, we have (x - 2)² + (y + 3)² = 3² = 9.
Therefore,
13 - p = 9 and solving for p gives us p = 13 - 9 = 4. The value of p is 4.