Write and equation for each translation of x^2+y^2=r^2 with the given radius. Left 3 units, up 2 units; radius 10

(x+3)^2 + (y-2)^2 = 100

show your work

Starting with the equation x^2+y^2=r^2, we can apply the following translations:

- Left 3 units: This means we need to add 3 to the x-coordinate so that the center of the circle is shifted to the left by 3 units. Therefore, x+3 replaces x in the equation.
- Up 2 units: This means we need to subtract 2 from the y-coordinate so that the center of the circle is shifted up by 2 units. Therefore, y-2 replaces y in the equation.
- Radius 10: This means r=10.

Putting it all together, we get:

(x+3)^2 + (y-2)^2 = 10^2

Expanding the left side, we get:

x^2 + 6x + 9 + y^2 - 4y + 4 = 100

Simplifying and rearranging, we get:

x^2 + y^2 + 6x - 4y + 13 = 100

Finally, we can write it in standard form:

(x+3)^2 + (y-2)^2 = 100

To translate the equation x^2 + y^2 = r^2 to the left by 3 units and up by 2 units, you need to subtract 3 from x and add 2 to y in the equation.

So, the equation for this translation would be:

(x - 3)^2 + (y + 2)^2 = r^2

Since the given radius is 10, the final equation would be:

(x - 3)^2 + (y + 2)^2 = 10^2