Example 3. For what value of $m$ does the line $y=-x+m$ intersect the ellipse $\frac{x^{2}}{20}+\frac{y^{2}}{5}=1$
(i) intersect;
(ii) be tangent,
(iii) be

separated? (Problem 13, page 132)
Solve the parametric equations of the ellipse as $\left\{\begin{array}{l}x=\sqrt{20} \cos \theta, \\ y=\sqrt{5} \sin \theta .\end{array}\right.$
When combined with the line $y-x+m$, we have $\sqrt{5} \sin \theta=-\sqrt{20} \cos \theta+m$, or $5 \sin (\theta+\phi)=m .(\phi=\operatorname{arctg} 2)$
Since the parameter (eccentric angle) $\theta \in[0,2 \pi$ ),
Therefore, when $\sin (\theta+\phi)= \pm 1$, i.e., $m= \pm 5$, $\theta$ has only one solution, at which time the line is tangent to the ellipse.

When $|\sin (\theta+\phi)|5$ or $m1$ is impossible, at which time the line is separated from the ellipse.

A similar method can be used to prove: "Prove that the condition for the line $y=k x+m$ to be tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is $k^{2} a^{2}+b^{2}=m^{2}$."