Example 1 Find the equation of the line $\iota$ passing through the intersection point of the two lines $x-2 y+4=0$ and $x+y-2=0$, and satisfying the following conditions.
(1) Passing through the point $(3,-2)$.
(2) Perpendicular to the line $3 x-4 y+7=0$.
Let the equation of the required line $l$ be $x-2 y+4+\lambda(x+y-2)=0$, i.e.,
$$
(\lambda+1) x+(\lambda-2) y+4-2 \lambda=0 .
$$
(1) Substituting $(3,-2)$, we get $3(\lambda+1)-2(\lambda-2)+4-2 \lambda=0 \cdot \lambda=-11$. Therefore, the equation of line $l$ is $10 x+13 y-26=0$.
(2) By the condition of perpendicularity, we have $3(\lambda+1)-4(\lambda-2)=0, \lambda=11$. The equation of the required line $l$ is $4 x+3 y-6=0$.