A common tangent T to the curves C 1 : x 2 4 + y 2 9 = 1 and C 2 : x 2 42 - y 2 143 = 1 does not pass…

Given curve $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$

Now let common tangents to the curves are

$T_1:y=mx\pm \sqrt{4m^2+9}$

and $T_2:y=mx\pm \sqrt{42m^2-143}$

Since they both are same,

So, $4m^2+9=42m^2-143$

$\Rightarrow 38 {\mathrm{m}}^2=152$

$\Rightarrow m=\pm 2$

So, $c=\pm \sqrt{4m^2+9}=\pm 5$

For given tangent not pass through $4^{\mathrm{th}}$ quadrant

So, $T:y=2x+5$

Now, comparing with point form of tangent  $\frac{x{x}_1}{4}+\frac{y{y}_1}{9}=1$

We get, $\frac{x_1}{8}=-\frac{1}{5}\Rightarrow x_1=-\frac{8}{5}$

And point form of tangent equation of hyperbola $\frac{x{x}_2}{42}-\frac{y{y}_2}{143}=1$ and $2x-y=-5$

We get $x_2=-\frac{84}{5}$

So $\left|2x_1+x_2\right|=\left|\frac{-100}{5}\right|=20$