The endpoints of the longest chord on a circle are (4, 5.5) and (4, 10.5).
the center of the circle is at the point ? and its radius is units ? the equation of this circle in standard form is ?

The longest chord on the circle is the diameter, and the center of the circle is the midpoint of the diameter.

Use the midpoint formula to find the center


the x-coordinates are the same, so you can just subtract the y-coordinates to find the radius. 
10.5 - 8 = 2.5

The equation of the circle is (x - h)² + (y - k)² = r² where r is the radius and (h, k) is the center.

(x - 4)² + (y - 8)² = 6.25