Radius of the largest circle which passes through the focus of the parabola $y^2=4x$ and contained in it, is

Let $r$ be the radius of the largest circle passing through the focus $\left(1,0\right)$ of $y^2=4x$

Clearly, centre of the circle will be on $X$-axis and its coordinates are $\left(1+r,0\right)$.

The equation of the circle, is

${\left(x-1-r\right)}^2+y^2=r^2$.

It touches $y^2=4x$. Therefore, the equation ${\left(x-r-1\right)}^2+4x=r^2$ must have equal roots.

$\therefore 4{\left(1-r\right)}^2-4\left(2r+1\right)=0$

$\Rightarrow r=4$

Hence, option (b) is correct.