The radius of the circle touching the parabola y² = x at (1, 1) and having the directrix of y² = x as its normal is

The equation of normal at P(1, 1) is $\text{y−1=−2(x−1) or y+2x=3 ...(i)}$
The directrix of the parabola ${\mathrm{y}}^2=\mathrm{x}\text{ is }\mathrm{x}=-\frac{1}{4} ...\left(\mathrm{ii}\right)$
The center of the circle is the intersection of two normals to the circle, i.e., (i) and (ii).
Therefore, the center is $\left(-\frac{1}{4},\frac{7}{2}\right)$.
Hence, the radius of the circle is $\sqrt{{\left(1+\frac{1}{4}\right)}^2+{\left(1-\frac{7}{2}\right)}^2}=\sqrt{\frac{25}{16}+\frac{25}{4}}=\frac{5\sqrt{5}}{4}$