## Problem Statement

Find the point of intersection of the line and the plane.

$$
\begin{aligned}
& \frac{x-1}{-2}=\frac{y-2}{1}=\frac{z+1}{-1} \\
& x-2 y+5 z+17=0
\end{aligned}
$$
## Solution

Let's write the parametric equations of the line.

$$
\begin{aligned}
& \frac{x-1}{-2}=\frac{y-2}{1}=\frac{z+1}{-1}=t \Rightarrow \\
& \left\{\begin{array}{l}
x=1-2 t \\
y=2+t \\
z=-1-t
\end{array}\right.
\end{aligned}
$$

Substitute into the equation of the plane:

$(1-2 t)-2(2+t)+5(-1-t)+17=0$

$1-2 t-4-2 t-5-5 t+17=0$

$-9 t+9=0$

$$
t=1
$$

Find the coordinates of the intersection point of the line and the plane:

$$
\left\{\begin{array}{l}
x=1-2 \cdot 1=-1 \\
y=2+1=3 \\
z=-1-1=-2
\end{array}\right.
$$

We get:

$(-1 ; 3 ;-2)$

## Problem Kuznetsov Analytic Geometry 14-7