Question

The equation (x-5)² + (y+2)²= 20 defines a circle that lies on the standard x,y coordinate plane. The line with endpoints (9,0) and (a,b) is a diameter of the circle. What is the value of a _______.

1) -13
2) -7
3) -5
4) -2
5) 1

Answer 1

The value of 'a' for the circle equation (x-5)² + (y+2)²= 20, with a diameter line endpoint at (9,0), is 1. This is found using the midpoint formula and considering the center of the circle as the midpoint of the diameter.

Step-by-step explanation:

The equation (x-5)² + (y+2)²= 20 defines a circle on the coordinate plane. The center of this circle is at (5,-2) and the radius is √20 which simplifies to 2√5. Since the line with endpoints (9,0) and (a,b) is a diameter, the other endpoint must be the opposite end of a diameter that passes through the center of the circle.

To find a, we use the midpoint formula which is the center of the circle. As the coordinates of the center are (5,-2) and one endpoint is (9,0), the other endpoint (a,b) must have the same distance from the center but in the opposite direction. Therefore, a = 5 - (9 - 5) = 5 - 4 = 1. Hence, the value of a is 1, which corresponds to option 5.