We are given a square $ABCD$ with center $O$ and direct orientation. We are also given some transformations. The problem asks for two things related to translation: a) Find the analytic expression of the translation by vector $\vec{AB}$. b) Find the Cartesian equation of the circle $(C')$, which is the image of the circle $(C)$ with equation $(x-1)^2 + (y+1)^2 = 2$ under the translation by vector $\vec{AB}$. The coordinate system is orthonormal with origin $O$, and basis vectors $\vec{OB}$ and $\vec{OA}$.

1. Problem Description

We are given a square $ABCD$ with center $O$ and direct orientation. We are also given some transformations. The problem asks for two things related to translation:

a) Find the analytic expression of the translation by vector $\vec{AB}$.

b) Find the Cartesian equation of the circle $(C')$, which is the image of the circle $(C)$ with equation $(x-1)^2 + (y+1)^2 = 2$ under the translation by vector $\vec{AB}$. The coordinate system is orthonormal with origin $O$, and basis vectors $\vec{OB}$ and $\vec{OA}$.

2. Solution Steps

a) Let $A = (x, y)$ be a point in the plane, and let $A' = (x', y')$ be its image under the translation by vector $\vec{AB}$. Then $\vec{AA'} = \vec{AB}$.

Since $ABCD$ is a square with center $O$, and the coordinate system is defined by $O$, $\vec{OB}$, and $\vec{OA}$, we can assume the coordinates of the vertices are $B=(1,0)$, $A=(0,1)$. Then $\vec{AB} = B - A = (1, 0) - (0, 1) = (1, -1)$.

The translation formula is:

$x' = x + 1$

$y' = y - 1$

Therefore, the analytic expression for the translation by $\vec{AB}$ is:

$x' = x + 1$

$y' = y - 1$

b) We are given the equation of circle $(C)$ as $(x-1)^2 + (y+1)^2 = 2$. We want to find the equation of circle $(C')$, which is the image of $(C)$ under the translation by $\vec{AB}$.

We have the translation formulas:

$x' = x + 1$

$y' = y - 1$

Then, $x = x' - 1$ and $y = y' + 1$.

Substituting these expressions into the equation of circle $(C)$:

$(x'-1-1)^2 + (y'+1+1)^2 = 2$

$(x'-2)^2 + (y'+2)^2 = 2$

Replacing $x'$ with $x$ and $y'$ with $y$, we get the equation of the circle $(C')$:

$(x-2)^2 + (y+2)^2 = 2$

3. Final Answer

a) The analytic expression for the translation by $\vec{AB}$ is:

$x' = x + 1$

$y' = y - 1$

b) The equation of the circle $(C')$ is:

$(x-2)^2 + (y+2)^2 = 2$