Find the value of K, if the line 4x + y + K = 0 is a tangent to ellipse x² + 3y² = 3.

Given ellipse ${\mathrm{x}}^2+3{\mathrm{y}}^2=3$
$\frac{{\mathrm{x}}^2}{3}+\frac{{\mathrm{y}}^2}{1}=1$
Given line 4x + y + K = 0
$\begin{array}{l}\Rightarrow \mathrm{y}=-4\mathrm{x}-\mathrm{k}\text{ is tangent to the ellipse }\frac{{\mathrm{x}}^2}{3}+\frac{{\mathrm{y}}^2}{1}=1\\ \mathrm{then} {\mathrm{c}}^2={\mathrm{a}}^2{\mathrm{m}}^2+{\mathrm{b}}^2\\ \Rightarrow {\mathrm{k}}^2=3\left(16\right)+1=49\\ \Rightarrow \mathrm{k}=\pm 7\end{array}$