Answer:
The equation of the circle is;
(x - [(-1))² + (y - [7])² = 8
Step-by-step explanation:
The given coordinates of the points P and Q are;
P = (-3, 5) and Q = (1, 9)
To find the equation of the circle that has segment PQ as a diameter
The equation of a circle is (x - h)² + (y - k)² = r²
Where;
(h, k) = The coordinates of the center of the circle
r = The radius of the circle
Given that segment PQ is the diameter of the circle, we have;
The center of the circle, O = The midpoint of PQ
The coordinates of the midpoint of segment PQ, O = ((-3 + 1)/2, (5 + 9)/2) = (-1, 7)
∴ The coordinates of the center of the circle, O = The coordinates of the midpoint of segment PQ = (-1, 7)
Therefore;
h = -1, k = 7
From PQ = The diameter of the required circle, we also have;
The length of PQ = 2 × (The radius of the circle) = 2 × r
The length of segment PQ = √((1 - (-3))² + (9 - 5)²) = √32 = 4·√2
Therefore;
The length of PQ = 4·√2 = 2 × r
r = (4·√2)/2 = 2·√2
r = 2·√2
r² = (2·√2)² = 8
r² = 8
The equation of the circle is therefore;
(x - h)² + (y - k)² = r²
(x - (-1))² + (y - 7)² = 8.
