a) The equation of the perpendicular bisector of the line segment AB is equal to y = - (2 / 3) · x + 3.
b) The equation of the circle is (x - 6)² + (y - 1)² = 34.
What are the equations of a perpendicular bisector of a line segment and a circle?
a) First, determine the midpoint of the line segment by using the midpoint formula:
M(x, y) = 0.5 · A(x, y) + 0.5 · B(x, y)
M(x, y) = 0.5 · (1, - 2) + 0.5 · (5, 4)
M(x, y) = (3, 1)
Second, calculate the slope of the line segment:
m = [4 - (-2)] / (5 - 1)
m = 6 / 4
m = 3 / 2
Third, derive the expression of the perpendicular bisector:
m' = - 1 / (3 / 2)
m' = - 2 / 3
b = y - m' · x
b = 1 - (- 2 / 3) · 3
b = 1 + 2
b = 3
The equation of the perpendicular bisector of the line segment AB is equal to y = - (2 / 3) · x + 3.
b) If the points A and B lie on a circle, then the following formula is constructed from Pythagorean Theorem:
(1 - 6)² + (- 2 - p)² = (5 - 6)² + (4 - p)²
25 + (4 + 4 · p + p²) = 1 + (16 - 8 · p + p²)
29 + 4 · p = 17 - 8 · p
12 · p = 12
p = 1
The coordinates of the center of the circle is (h, k) = (6, 1).
The radius of the circle is:
r = √[(1 - 6)² + (- 2 - 1)²]
r = √[(- 5)² + (- 3)²]
r = √34
The equation of the circle is (x - 6)² + (y - 1)² = 34.
To learn more on equations of circles: https://brainly.com/question/10165274
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