# How do you describe the translation of circle (x - 1)2 + (y - 4)2 = 25 that results in an image whose equation is (x + 1)2 + (y - 5)2 = 25?

Mar 23, 2016

$\left(\begin{matrix}- 2 \\ 1\end{matrix}\right)$

#### Explanation:

Compare the given equations to the standard equation.

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

where (a,b) are the coords of centre and r , is the radius.

so ${\left(x - 1\right)}^{2} + {\left(y - 4\right)}^{2} = 25 \text{ has centre (1,4)and r =5}$

and ${\left(x + 1\right)}^{2} + {\left(y - 5\right)}^{2} = 25 \text{ has centre (-1,5) and r = 5 }$

Under a translation the circle retains it's shape, but position is changed.
Just need to consider how the centres have changed.

that is : (1,4) → (-1,5) and as a translation $\left(\begin{matrix}- 2 \\ 1\end{matrix}\right)$