lunar eclipse - Does earth's Umbra reach Sun-Earth L2?

The Lagrangian point $L_2$ is very close to the most distant point from Earth with an umbra.
$L_2$ is like the radius of the Hill sphere at $r=asqrt[3]{frac{m}{3M}}$ for circular orbits, with $m$ the mass of Earth, $M$ the mass of the Sun, and $a$ the distance Earth-Sun. The ratio $frac{m}{3M}$ of the Earth and the triple mass of the Sun is almost exactly $10^{-6}$, the cubic root hence $0.01$.

The diameter ratio of Earth and Sun is about $1/109$. Therefore the umbra of Earth ends near $92%$ the distance to $L_2$.

The answer to another bonus question would then be: If Earth would be $9%$ larger in diameter, but with the same mass, its umbra would end almost exactly at $L_2$.

Earth's orbit isn't perfectly circular, but the aphel/perihel ratio of about $1.04$ is insufficient to question the result qualitatively.
The error of the implicite assumptions $tan x=x=sin x$ is negligible at the considered level of accuracy.