Equation of family of circle touching y-axis at $\left(0, 4\right)$ is given by $\left(x - 0\right)^{2} + \left(y - 4\right)^{2} + \lambda x = 0.$
$\because$ It passes through $\left(2, 0\right)$
$\Rightarrow \lambda=-10$
$\Rightarrow $ Required circle is $\left(x - 0\right)^{2} + \left(y - 4\right)^{2} - 10x = 0$
$\Rightarrow x^{2}+y^{2}-10x-8y+16=0$
center of circle $\equiv \left(5, 4\right)$ and radius $= 5$ distance of $4x + 3y - 8 = 0$ from $\left(5, 4\right)$
$=\left|\frac{24}{5}\right|\ne$ radius