I have to express the following problem as a semidefinite program
$$\begin{array}{ll} \text{minimize} & F(x,y) := x + y +1\\ \text{subject to} & (x-1)^2+y^2 \leq 1 \tag{1}\end{array}$$
Only affine equality conditions should be used. The hint was to examine the structure of $\mathbb{S}^2_+$, the cone of symmetric positive semidefinite matrices.
The characteristic polynomial of such a matrix is
$$C=\lambda^2-(a_{11}+a_{22})\lambda - a_{12}a_{21}$$
which has a similar form to the rewritten condition (1) $x^2-2x+y^2\leq0$. If $a_{11}=a_{22}=1,\lambda = x$ and $a_{12}=-a_{21}=y$ the characteristic polynomial would be $x^2-2x+y^2=0$. Is this useful?
My problem is, that I have no clue how to formulate a $\leq$ with equality conditions.