8,9

Given a point $M(x ; y)$. Find the coordinates of the point symmetric to point $M$ with respect to a) the origin; b) the point $K(a ; b)$.

#
Let the point $M^{\prime}\left(x^{\prime}\right.$; $\left.y^{\prime}\right)$ be symmetric to the point $M(x ; y)$ with respect to the origin, i.e., the point $O(0 ; 0)$. Since $O$ is the midpoint of the segment $M M^{\prime}$, and the coordinates of the midpoint of a segment are the arithmetic means of the corresponding coordinates of its endpoints, then

$$
0=\frac{x+x^{\prime}}{2}, 0=\frac{y+y^{\prime}}{2}
$$

Therefore, $x^{\prime}=-x, y_{1}=-y$.

Let the point $M^{\prime}\left(x^{\prime} ; y^{\prime}\right)$ be symmetric to the point $M(x ; y)$ with respect to the point $Q(a ; b)$. Since $Q$ is the midpoint of the segment $M M^{\prime}$,

$$
a=\frac{x+x^{\prime}}{2}, b=\frac{y+y^{\prime}}{2}
$$

Therefore, $x^{\prime}=2 a-x, y_{1}=2 b-y$.

## Answer

a) $(-x ;-y) ;$ b) $(2 a-x ; 2 b-y)$.