A common tangent T to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$  and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If T touches $C_1$  at $(x_1,y_1)$  and $C_2$ at $(x_2,y_2).$ Then $|2x_1+x_2|$  is equal to

Let common lengths are  $T_1=y=mx\pm \sqrt{4m^2+9}$
                                              $T_2=y=mx\pm \sqrt{42m^2-143}$
   $\begin{array}{l}so4m^2+9=42m^2-143\\ 38m^2=152\Rightarrow m=\pm 2\end{array}$
$f_0$ given tangent not pass through 4th quadrant
$\therefore T=y=2x+5$
now comparing with $\frac{x{x}_1}{4}+\frac{y{y}_1}{9}=1,\frac{x{x}_2}{42}-\frac{y{y}_2}{143}=1$
we get  $\Rightarrow x_1=-8/5$
$\therefore x_2=-84/5$
This  $|2x_1+x_2|=20$