Answer:a) To verify if points P(5, 7) and Q(7, -5) lie on the circle with equation x² + y² = 74, we can substitute the x and y coordinates of each point into the equation and check if it holds true.
For point P(5, 7):
5² + 7² = 25 + 49 = 74
The sum of the squares of the x and y coordinates equals 74, which is the same as the right side of the equation. Therefore, point P lies on the circle.
For point Q(7, -5):
7² + (-5)² = 49 + 25 = 74
The sum of the squares of the x and y coordinates also equals 74, confirming that point Q lies on the circle.
b) To verify if the right bisector of chord PQ passes through the center of the circle, we need to determine the midpoint of chord PQ and compare it to the center of the circle.
Midpoint of chord PQ = [(x₁ + x₂) / 2, (y₁ + y₂) / 2]
where (x₁, y₁) are the coordinates of point P and (x₂, y₂) are the coordinates of point Q.
Using the coordinates of P(5, 7) and Q(7, -5):
Midpoint of PQ = [(5 + 7) / 2, (7 + (-5)) / 2]
= [12 / 2, 2 / 2]
= [6, 1]
The midpoint of PQ is (6, 1).
The center of the circle is at the origin (0, 0) since the equation of the circle is x² + y² = 74, which has a radius of √74.
Since the midpoint of PQ (6, 1) is not the same as the center of the circle (0, 0), the right bisector of chord PQ does not pass through the center of the circle.
Explanation: