= Solution
Normalize the specified direction to $\hat{\mathbf u}=(1,1)/\sqrt2$. At distance $s\geq0$ the squared radius is $(1+s/\sqrt2)^2+(2+s/\sqrt2)^2=5+3\sqrt2s+s^2$. The exact change $\Delta T=0.12$ therefore requires $5+3\sqrt2s+s^2=5e^{0.12}$. The positive root is
$$
\boxed{s=\frac{\sqrt{20e^{0.12}-2}-3\sqrt2}{2}.}
$$
For comparison, the initial <directional derivative> is $(2/5,4/5)\cdot(1,1)/\sqrt2=3\sqrt2/5$. Its first-order distance estimate is
$$
\boxed{s\simeq\frac{0.12}{3\sqrt2/5}=\frac{\sqrt2}{10}.}
$$
These give approximately $0.145$ and $0.141$. The exact expression accounts for the changing gradient along a finite straight displacement.
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