Example 2. Form the equations of the normal and the tangent plane to the sphere $x^{2}+y^{2}+z^{2}-2 x+4 y-6 z+5=0$ at the point $M_{0}(3,-1,5)$.
Solution. Let's find the partial derivatives of the function $F(x, y, z) = x^{2} + y^{2} + z^{2} - 2x + 4y - 6z + 5$:

$$
\frac{\partial F}{\partial x} = 2x - 2, \frac{\partial F}{\partial y} = 2y + 4, \frac{\partial F}{\partial z} = 2z - 6
$$

Evaluating these derivatives at the point $M_{0}(3, -1, 5)$, we get the direction vector of the normal $\bar{m} = \{4, 2, 4\}$.

According to formula (8.3), we form the equation of the tangent plane

$$
4(x - 3) + 2(y + 1) + 4(z - 5) = 0
$$

or

$$
2x + y + 2z - 15 = 0
$$

Based on formula (8.4), we find the equations of the normal

$$
\frac{x - 3}{2} = \frac{y + 1}{1} = \frac{z - 5}{2}
$$