Tangent Distances to the Foci

The ellipse $\Gamma:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has foci $F_1(-c,0)$ and $F_2(c,0)$. A line $l$ is tangent to $\Gamma$, and $d_1$, $d_2$ denote the distances from $F_1$, $F_2$ to $l$.

1. Prove that $d_1\cdot d_2=b^2$. 2. Prove that $2b\leqslant d_1+d_2\leqslant 2a$. 3. Prove that $\dfrac{a-c}{a+c}\leqslant\dfrac{d_1}{d_2}\leqslant\dfrac{a+c}{a-c}$.