Let, the lengths of the semi-major axis and semi-minor axis of the ellipse are $a$ & $b$ respectively, then the distance of the focus $\left(\mathrm{3,4}\right)$ from the corresponding directrix $\left(x+y-1=0\right)$ is equal to $\frac{a}{e}-ae$
$\Rightarrow \frac{a}{\frac{1}{2}}-a\left(\frac{1}{2}\right)=\left|\frac{3+4-1}{\sqrt{2}}\right|=3\sqrt{2}$
$\Rightarrow \frac{3a}{2}=3\sqrt{2}\Rightarrow a=2\sqrt{2}$
Distance between the focus and centre $=ae=\sqrt{2}$
The slope of the axis of the ellipse is $1=\tan \theta$
$\Rightarrow \left(\cos \theta ,\sin \theta \right)=\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ or $\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$
The points on the axis of the ellipse at a distance $\sqrt{2}$ units from $\left(\mathrm{3,4}\right)$ are $\left(3\pm \sqrt{2}\left(\frac{1}{\sqrt{2}}\right),4\pm \sqrt{2}\left(\frac{1}{\sqrt{2}}\right)\right)$
$\Rightarrow$ Points are $\left(\mathrm{2,3}\right)$ or $\left(\mathrm{4,5}\right)$ , but the centre is $\left(\mathrm{4,5}\right)$ because it is far from the directrix as compared to $\left(\mathrm{2,3}\right)$.