To find the center and radius of a circle given the endpoints of its diameter, we can follow these steps:
Step 1: Find the center of the circle.
The center of the circle is the midpoint of the diameter. We can find the midpoint between points P(x1, y1) and Q(x2, y2) using the midpoint formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)For our points P(10, 2) and Q(46, y2), we substitute:
Center = ((10 + 46) / 2, (2 + y2) / 2)Now, we need the y-coordinate of point Q. However, since we are finding the center based solely on the x-coordinates:
Center = (28, (2 + y2) / 2)To solve for the center, we only need the x-coordinate, which is 28, and for a complete solution we must assume y2 is known or provided.
Step 2: Find the radius of the circle.
The radius is half the length of the diameter. We first need the length of the diameter, which can be calculated using the distance formula:
Distance = √[(x2 - x1)² + (y2 - y1)²]Substituting the coordinates of points P(10, 2) and Q(46, y2):
Diameter = √[(46 - 10)² + (y2 - 2)²]This simplifies to:
Diameter = √[36² + (y2 - 2)²] = √[1296 + (y2 - 2)²]To find the radius:
Radius = Diameter / 2 = ½ √[1296 + (y2 - 2)²]In conclusion, the center of the circle in terms of the y-coordinate of point Q is at (28, (2 + y2) / 2), and the radius can be calculated based on the known y-coordinate of point Q.