[tex](x-4)^2+(y-2)^2=7^2[/tex]
To know if a point lies on a circle you use the (x,y) of each point in the equation and prove it that correspond to a mathematical congruence.
A. (- 1 , 4)
[tex]\begin{gathered} (-1-4)^2+(4-2)^2=7^2 \\ -5^2+2^2=49 \\ 25+4=49 \\ 29=49 \end{gathered}[/tex]
As 29 is not equal to 49, this point doesn't lie in the circle
B. ( 8 , 3)
[tex]\begin{gathered} (8-4)^2+(3-2)^2=7^2 \\ 4^2+1^2=49 \\ 16+1=49 \\ 17=49 \end{gathered}[/tex]
As 17 is not equal to 49, this point doesn't lie in the circle
C. (9 , 0)
[tex]\begin{gathered} (9-4)^2+(0-2)^2=7^2 \\ 5^2+(-2)^2=49 \\ 25+4=49 \\ 29=49 \end{gathered}[/tex]
As 29 is not equal to 49, this point doesn't lie in the circle
D. (-2 , 2)
[tex]\begin{gathered} (-2-4)^2+(2-2)^2=7^2 \\ -6^2+0^2=49 \\ 36=49 \end{gathered}[/tex]
As 36 is not equal to 49, this point doesn't lie in the circle
None of the points lies on the circle.
The next graph represents the circle and the 4 given points:
