Example 1. Find the coordinates of the intersection point of the two tangent lines to the ellipse at the points of intersection with the line $x+4 y-2=0$ and the ellipse $\frac{x^{2}}{2}$ $+\mathrm{y}^{2}=1$.
Let $\mathrm{P}^{\prime}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be the point sought. Then the chord of contact of the circle with respect to $P^{\prime}$ is given by $\frac{x^{\prime} x}{2}+y^{\prime} y-1=0$, which should coincide with the line equation $\mathrm{x}+4 \mathrm{y}-2=0$. Therefore, we have $\frac{\mathrm{x}^{\prime}}{1}=\frac{\mathrm{y}^{\prime}}{4}=\frac{-1}{-2}$. Solving this, we get $x^{\prime}=1, y^{\prime}=2$, so the intersection point is $\mathrm{P}^{\prime}(1,2)$.