As $y-\mathrm{axis}$ is tangent to the circle therefore the centre of the circle would be $\left(x,2\right)$ and distance of centre from $\left(0,2\right)$ and $\left(-2,4\right)$ should be equal.

$\therefore$ $\sqrt{h^2+0^2}=$$\sqrt{{\left(h+2\right)}^2+2^2}$

$\Rightarrow$$h^2=h^2+4h+4+4$

$\Rightarrow$$h=-2$

Hence, centre is $\left(-2,2\right)$.

$\therefore$ $2x-3y+10=0$ is the diameter, as centre lies on it.