The length of the chord of the parabola y² = x which is bisected at the point (2, 1) is

Chord through (2, 1) is
$\frac{\mathrm{x}-2}{\cos \mathrm{\theta }}=\frac{\mathrm{y}-1}{\sin \mathrm{\theta }}=\mathrm{r}$………(1)
Solving (i) with the parabola
y2 = x, we have
$(1+\mathrm{rsin}\mathrm{\theta }{)}^2=2+\mathrm{rcos}\mathrm{\theta }$
or ${\sin }^2{\mathrm{\theta r}}^2+(2\sin \mathrm{\theta }-\cos \mathrm{\theta })\mathrm{r}-1=0$
This equation has two roots: ${\mathrm{r}}_1=\mathrm{AC}\text{ and }{\mathrm{r}}_2=-\mathrm{BC}$ Then,
Sum of roots, r₁ + r₂ = 0
or $2\sin \mathrm{\theta }-\cos \mathrm{\theta }=0\text{ or }\tan \mathrm{\theta }=\frac{1}{2}$
$\begin{array}{r}\mathrm{AB}=\left|{\mathrm{r}}_1-{\mathrm{r}}_2\right|=\sqrt{{\left({\mathrm{r}}_1+{\mathrm{r}}_2\right)}^2-4{\mathrm{r}}_1{\mathrm{r}}_2}\\ =\sqrt{4\frac{1}{{\sin }^2\mathrm{\theta }}}=2\sqrt{5}\end{array}$