Question

Which translation results in a circle whose center is at the origin? Question 5 options: 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. 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. 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. 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.

Answer 1

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

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

`(x+2)^(2)+(y-6)^(2)=4`

The center of the circle is the point
`(-2,6)`

The rule of the translation is

`(x,y)-------> (x+2,y-6)`

Applying the rule to the center

`(-2,6)-------> (-2+2,6-6)`

`(-2,6)-------> (0,0)`

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

`(x-1)^(2)+(y+2)^(2)=4`

The center of the circle is the point
`(1,-2)`

The rule of the translation is

`(x,y)-------> (x-1,y-2)`

Applying the rule to the center

`(1,-2)-------> (1-1,-2-2)`

`(1,-2)-------> (0,-4)`

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

`(x-2)^(2)+(y+3)^(2)=4`

The center of the circle is the point
`(2,-3)`

The rule of the translation is

`(x,y)-------> (x+2,y+3)`

Applying the rule to the center

`(2,-3)-------> (2+2,-3+3)`

`(2,-3)-------> (4,0)`

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

`(x-3)^(2)+(y-5)^(2)=4`

The center of the circle is the point
`(3,5)`

The rule of the translation is

`(x,y)-------> (x-3,y+5)`

Applying the rule to the center

`(3,5)-------> (3-3,5+5)`

`(3,5)-------> (0,10)`

Therefore

The translation results in a circle whose center is not at the origin