Final answer:
To find the circle's equation, the center is calculated as the midpoint of the diameter, C = (1, -0.5), and the radius is half the distance between P and Q, r = √205/2. Using the standard circle equation and substituting in these values, the circle's equation is (x - 1)² + (y + 0.5)² = 205/4.
Step-by-step explanation:
The circle's equation can be found by determining the center and the radius length. The midpoint of the diameter from point P(-2,6) to point Q(4,-7) gives the circle's center (C), and the distance between P and Q will provide the diameter, which can be used to find the radius (r).
To find the midpoint (C) use the formula:
- C = ((x1 + x2)/2, (y1 + y2)/2)
For points P and Q, this calculation is:
- C = ((-2 + 4)/2, (6 - 7)/2)
- C = (1, -0.5)
To find the radius, calculate the distance between P and Q, which is the length of diameter d:
- d = √((x2 - x1)² + (y2 - y1)²)
- d = √((4 + 2)² + (-7 - 6)²)
- d = √(6² + 13²) = √(36 + 169) = √205
Since the radius r is half of the diameter, r = √205/2.
The general form of the circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. Substituting h = 1, k = -0.5, and r = √205/2, the equation of the circle is:
- (x - 1)² + (y + 0.5)² = ( √205/2 )²
- (x - 1)² + (y + 0.5)² = 205/4