Answer

e. (x - 3)^2 + (y - 2)^2 = 9f. 3g. 6h. No

Explanation

e. The equation of a circle in the coordinate plane is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius. In this case, the center of the circle is at the point (3,2), so h = 3 and k = 2. A spoke of the circle connects the center to a point on the rim at (6,2), which means the radius of the circle is the distance between these two points. The distance between two points (x_1, y_1) and (x_2, y_2) in the coordinate plane is given by \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. So, the radius r is \sqrt{(6 - 3)^2 + (2 - 2)^2} = 3. Substituting h = 3, k = 2, and r = 3 into the equation of the circle, we get (x - 3)^2 + (y - 2)^2 = 3^2, which simplifies to (x - 3)^2 + (y - 2)^2 = 9.f. As calculated above, the radius of the circle is 3.g. The diameter of a circle is twice its radius. So, the diameter of this circle is 2 * 3 = 6.h. The point (4,-2) lies inside the circle if the distance from this point to the center of the circle is less than the radius. The distance from (4,-2) to (3,2) is \sqrt{(4 - 3)^2 + (-2 - 2)^2} = \sqrt{17}, which is greater than the radius of the circle (3). Therefore, the point (4,-2) does not lie inside the circle.