Given:
The center of the circle = (-1,0)
Point on the circle = (2,-4).
To find:
The equation of the circle.
Solution:
Distance between two points is
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Distance between the center (-1,0) and the point on the circle (2,-4) is the radius of the circle.
[tex]r=\sqrt{(2-(-1))^2+(-4-0)^2}[/tex]
[tex]r=\sqrt{(2+1)^2+(-4)^2}[/tex]
[tex]r=\sqrt{(3)^2+(-4)^2}[/tex]
On further simplification, we get
[tex]r=\sqrt{9+16}[/tex]
[tex]r=\sqrt{25}[/tex]
[tex]r=5[/tex]
Now, the equation of the circle is
[tex](x-h)^2+(y-k)^2=r^2[/tex]
Where, (h,k) is the center and r is the radius of the circle.
The radius of the circle is 5 and the center is at (-1,0), so the equation of the circle is
[tex](x-(-1))^2+(y-0)^2=5^2[/tex]
[tex](x+1)^2+y^2=25[/tex]
Therefore, the first option is correct, i.e., [tex](x+1)^2+y^2=25[/tex].
