Answer :
The equation of the circle is defined by:
[tex](x-2)^2+y^2=9[/tex]In order to determine the points that lie on the circle,
First, we need to substitute the value of x into the circle equation and solve for y. If the value of y obtained corresponds with the y-coordinate for that option, then we have our answers.
For [option a] with coordinates (-1, 0)
Put x = -1 into the circle equation
[tex]\begin{gathered} (-1-2)^2+y^2=9 \\ (-3)^2+y^2=9 \\ 9+y^2=9 \\ y^2=9-9=0 \\ y=0 \end{gathered}[/tex]For [option b] with coordinates (5, 0)
Put x = 5 into the circle equation
[tex]\begin{gathered} (5-2)^2+y^2=9 \\ 3^2+y^2=9 \\ 9+y^2=9 \\ y^2=9-9=0 \\ y=0 \end{gathered}[/tex]For [option c] with coordinates (0,√5)
Put x = 0 into the circle equation
[tex]\begin{gathered} (0-2)^2+y^2=9 \\ (-2)^2+y^2=9 \\ 4+y^2=9 \\ y^2=9-4=5 \\ y=\sqrt[]{5} \end{gathered}[/tex]For [option d] with coordinates (3, √5)
Put x = 3 into the circle equation
[tex]\begin{gathered} (3-2)^2+y^2=9 \\ 1^2+y^2=9 \\ 1+y^2=9 \\ y^2=9-1 \\ y^2=8 \\ y=\sqrt[]{8} \end{gathered}[/tex]Therefore, from our calculations, the options a, b, and c only are points on the circle because they satisfy the condition of the given equation of the circle.
