Answer:
case 1) Circle open parentheses x plus 2 close parentheses squared plus open parentheses y minus 6 close parentheses squared equals 4 translated 2 units right and 6 units down
Step-by-step explanation:
Verify the 4 Options
case 1) Circle open parentheses x plus 2 close parentheses squared plus open parentheses y minus 6 close parentheses squared equals 4 translated 2 units right and 6 units down
The equation of the circle is
[tex](x+2)^{2}+(y-6)^{2}=4[/tex]
The center of the circle is the point [tex](-2,6)[/tex]
The rule of the translation is
[tex](x,y)-------> (x+2,y-6)[/tex]
Applying the rule to the center
[tex](-2,6)-------> (-2+2,6-6)[/tex]
[tex](-2,6)-------> (0,0)[/tex]
Therefore
The translation results in a circle whose center is at the origin
case 2) Circle open parentheses x minus 1 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 4 translated 1 unit left and 2 units down
The equation of the circle is
[tex](x-1)^{2}+(y+2)^{2}=4[/tex]
The center of the circle is the point [tex](1,-2)[/tex]
The rule of the translation is
[tex](x,y)-------> (x-1,y-2)[/tex]
Applying the rule to the center
[tex](1,-2)-------> (1-1,-2-2)[/tex]
[tex](1,-2)-------> (0,-4)[/tex]
Therefore
The translation results in a circle whose center is not at the origin
case 3) Circle open parentheses x minus 2 close parentheses squared plus open parentheses y plus 3 close parentheses squared equals 4 translated 2 units right and 3 units up
The equation of the circle is
[tex](x-2)^{2}+(y+3)^{2}=4[/tex]
The center of the circle is the point [tex](2,-3)[/tex]
The rule of the translation is
[tex](x,y)-------> (x+2,y+3)[/tex]
Applying the rule to the center
[tex](2,-3)-------> (2+2,-3+3)[/tex]
[tex](2,-3)-------> (4,0)[/tex]
Therefore
The translation results in a circle whose center is not at the origin
case 4) Circle open parentheses x minus 3 close parentheses squared plus open parentheses y minus 5 close parentheses squared equals 4 translated 3 units left and 5 units up
The equation of the circle is
[tex](x-3)^{2}+(y-5)^{2}=4[/tex]
The center of the circle is the point [tex](3,5)[/tex]
The rule of the translation is
[tex](x,y)-------> (x-3,y+5)[/tex]
Applying the rule to the center
[tex](3,5)-------> (3-3,5+5)[/tex]
[tex](3,5)-------> (0,10)[/tex]
Therefore
The translation results in a circle whose center is not at the origin
