10,11

On the plane, points $A_{\mathrm{t}}=(1+t, 1+t)$ and $B_{\mathrm{t}}=(-1+t, 1-t)$ are given. Describe the set swept by all lines $A_{\mathrm{t}} B_{\mathrm{t}}$ for all real numbers $t$.
The line $A_{t} B_{\mathrm{t}}$ is given by the equation

$$
\frac{x-1-t}{y-1-t}=\frac{1+t+1-t}{1+t-1+t}=\frac{1}{t}
$$

i.e.,

$$
y=1-t^{2}+t x=-\left(t-\frac{x}{2}\right)^{2}+\frac{x^{2}}{4}+1
$$

For a fixed $x$, as $t$ runs through all real values, $y$ takes on all values not exceeding $\left(x^{2} / 4\right)+1$. Thus, the desired set is given by the inequality $y \leqslant\left(x^{2} / 4\right)+1$.