Answer:
![\rm\displaystyle (x - 4) ^{2} + {(y + 4)}^{2} = 36](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20%28x%20-%204%29%20%5E%7B2%7D%20%20%2B%20%20%7B%28y%20%20%2B%204%29%7D%5E%7B2%7D%20%20%3D%20%2036)
Step-by-step explanation:
the given points are the diameter points of circle because notice that in the both points y coordinate is the same therefore it's a horizontal diameter
since (10,-4),(-2,-4) are the diameter points of the circle the midpoint of the diameter will be the centre of the circle
remember midpoint formula,
![\displaystyle M = \left( \frac{x _{1} + x_{2} }{2} , \frac{ y_{2} + y_{2}}{2} \right)](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20M%20%3D%20%20%5Cleft%28%20%5Cfrac%7Bx%20_%7B1%7D%20%2B%20%20x_%7B2%7D%20%7D%7B2%7D%20%2C%20%5Cfrac%7B%20y_%7B2%7D%20%2B%20%20y_%7B2%7D%7D%7B2%7D%20%20%20%5Cright%29)
let,
thus substitute:
![\rm\displaystyle M = \left( \frac{10 + ( - 2)}{2} , \frac{ - 4 + ( - 4)}{2} \right)](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20M%20%3D%20%20%5Cleft%28%20%5Cfrac%7B10%20%2B%20%20%28%20-%202%29%7D%7B2%7D%20%2C%20%5Cfrac%7B%20-%204%20%2B%20%20%28%20-%204%29%7D%7B2%7D%20%20%20%5Cright%29)
simplify addition:
![\rm\displaystyle M = \left( \frac{8}{2} , \frac{ - 8}{2} \right)](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20M%20%3D%20%20%5Cleft%28%20%5Cfrac%7B8%7D%7B2%7D%20%2C%20%5Cfrac%7B%20-%208%7D%7B2%7D%20%20%20%5Cright%29)
simplify division:
![\rm\displaystyle M = \left( 4, - 4 \right)](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20M%20%3D%20%20%5Cleft%28%204%2C%20%20-%204%20%5Cright%29)
so the centre of the circle is <u>(</u><u>4</u><u>,</u><u>-</u><u>4</u><u>)</u>
since it's a horizontal diameter the the redious will be the difference between the x coordinate of the Midpoint and the any x coordinate of the given two points but I'll use (-2,-4) therefore the redious is
![\displaystyle r = 4 - ( - 2)](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20r%20%20%3D%204%20-%20%28%20-%202%29)
simplify which yields:
![\displaystyle\boxed{ r =6}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%5Cboxed%7B%20r%20%20%3D6%7D)
recall the equation of circle
![\displaystyle (x - h) ^{2} + {(y - k)}^{2} = {r}^{2}](https://tex.z-dn.net/?f=%20%5Cdisplaystyle%20%28x%20-%20h%29%20%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20k%29%7D%5E%7B2%7D%20%20%3D%20%20%7Br%7D%5E%7B2%7D%20)
we acquire that,
therefore substitute:
![\rm\displaystyle (x - 4) ^{2} + {(y - ( - 4))}^{2} = {6}^{2}](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20%28x%20-%204%29%20%5E%7B2%7D%20%20%2B%20%20%7B%28y%20-%20%28%20-%204%29%29%7D%5E%7B2%7D%20%20%3D%20%20%7B6%7D%5E%7B2%7D%20)
simplify:
![\rm\displaystyle (x - 4) ^{2} + {(y + 4)}^{2} = 36](https://tex.z-dn.net/?f=%20%20%5Crm%5Cdisplaystyle%20%28x%20-%204%29%20%5E%7B2%7D%20%20%2B%20%20%7B%28y%20%20%2B%204%29%7D%5E%7B2%7D%20%20%3D%20%2036)
and we are done!
also refer the attachment
(the graph is web resource of desmos)