Answer:
1) Use translation to coincide the center of a circle with the center of the other circle.
2) We construct the respective loci for the circles.
3) Compare each loci by direct inspection ([tex]f(x,y)[/tex] must be the same and radii must be different but constant) to conclude the similarity of the two circles.
Step-by-step explanation:
Geometrically speaking, a circle is formed after knowing its center and radius.
1) Use translation to coincide the center of a circle with the center of the other circle. In this case, we translate the center ([tex]C_{A} (x,y) = (6,7)[/tex]) of the circle A to the location of the center of the circle B ([tex]C_{B}(x,y) = (2,4)[/tex]):
[tex]C_{A'} (x,y) = (6,7) + [(2,4) - (6,7)][/tex]
[tex]C_{A'} (x,y) = (2,4)[/tex]
2) We construct the respective loci for the circles.
By Analytical Geometric, a circle is represented by the following locus:
[tex](x-h)^{2}+(y-k)^{2} = r^{2}[/tex]
Where:
[tex]x, y[/tex] - Coordinates of a point of the circunference.
[tex]h, k[/tex] - Coordinates of the center of the circle.
[tex]r[/tex] - Radius of the circle.
3) Compare each loci by direct inspection ([tex]f(x,y)[/tex] must be the same and radii must be different but constant) to conclude the similarity of the two circles.
Then, the circles A' and B are represented by the following loci:
Circle A'
[tex](x-2)^{2} + (y-4)^{2} = 16[/tex]
Circle B
[tex](x-2)^{2} + (y-4)^{2} = 256[/tex]
Since both the [tex]f(x,y)[/tex] component of each circle is the same and radii are different but constant, then we conclude that circle A is similar to circle B.
