Example 5 Find the equation of the curve $E^{\prime}$ symmetric to the curve $E: 5 x^{2}+12 x y-22 x-12 y-$ $19=0$ with respect to the line $l: x-y+2=0$.
Solution: Given $l: x=y-2, y=x+2$, substituting into equation $E$ yields $5(y-2)^{2}+12(y-2)(x+2)-22(y-2)-$ $12(x+2)-19=0$, simplifying to $12 x y+5 y^{2}-36 x-$ $18 y-27=0$, which is the desired result.