Answer:
[tex]\begin{gathered} (A)\Rightarrow\text{ Point B lies inside the circle Because:} \\ CA=\sqrt[]{34} \\ CB=4 \end{gathered}[/tex]
Explanation: The general form of the equation of the circle is as follows:
[tex]\begin{gathered} (x-x_o)^2+(y-y_o)^2=k^2 \\ (x_o,y_o)\rightarrow\text{ Center of the circle}\rightarrow(4,-2) \\ \therefore\Rightarrow \\ (x-4)^2+(y+2_{})^2=k^2\Rightarrow(1) \end{gathered}[/tex]
(1) is the equation of the circle. since it passes through the point (1,3), therefore substituting the (x,y) values of it in (1) gives the value complete equation (1) as follows:
[tex]\begin{gathered} (1-4)^2+(3+2_{})^2=k^2 \\ 9+25=k^2 \\ k^2=34 \\ \therefore\Rightarrow \\ (x-4)^2+(y+2_{})^2=k^2\Rightarrow(2) \end{gathered}[/tex]
(2) is the complete equation for the circle, the following graph shows if point B is inside or outside the circle.
