How do you find the equation that represents the image of circle (x- 5)^2 + (y + 12)^2 = 169 after a translation 2 units right and 3 units down?

Jan 30, 2016

Adjust to standard form to see what the original centre is, translate the centre, then plug back into the equation to find:

${\left(x - 7\right)}^{2} + {\left(y - \left(- 15\right)\right)}^{2} = {13}^{2}$

or:

${\left(x - 7\right)}^{2} + {\left(y + 15\right)}^{2} = 169$

Explanation:

The standard equation of a circle with centre $\left(h , k\right)$ and radius $r$ is:

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {r}^{2}$

With minor changes our starting equation is:

${\left(x - 5\right)}^{2} + {\left(y - \left(- 12\right)\right)}^{2} = {13}^{2}$

That is: It is a circle with centre $\left(5 , - 12\right)$ and radius $13$.

If translated $2$ units right and $3$ units down then the centre will be $\left(5 + 2 , - 12 - 3\right) = \left(7 , - 15\right)$ and the equation becomes:

${\left(x - 7\right)}^{2} + {\left(y - \left(- 15\right)\right)}^{2} = {13}^{2}$

or putting it back in similar form to the starting equation:

${\left(x - 7\right)}^{2} + {\left(y + 15\right)}^{2} = 169$