Let the locus of the mid points of the chords of circle x ^ 2 + (y...

Let the mid point of the chords be $\left({x}_{1}, {y}_{1}\right)$.

The given equation of circle is ${x}^{2} + {\left(y - 1\right)}^{2} = 1$.

$⇒ {x}^{2} + {y}^{2} - 2y = 0$

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

This locus passes through the origin,

$⇒ 0 + 0 - \left(0 + {y}_{1}\right) = {{x}_{1}}^{2} + {{y}_{1}}^{2} - 2{y}_{1}$

$⇒ - {y}_{1} = {{x}_{1}}^{2} + {{y}_{1}}^{2} - 2{y}_{1}$

$⇒ {{x}_{1}}^{2} + {{y}_{1}}^{2} - {y}_{1} = 0$

So, the locus is ${x}^{2} + {y}^{2} - y = 0$

It intersects with the line $x + y = 1$.

$⇒ {\left(1 - y\right)}^{2} + {y}^{2} - y = 0$

$⇒ 1 + {y}^{2} - 2y + {y}^{2} - y = 0$

$⇒ 2{y}^{2} - 2y - y + 1 = 0$

$⇒ 2y\left(y - 1\right) - \left(y - 1\right) = 0$

$⇒ \left(2y - 1\right)\left(y - 1\right) = 0$

$⇒ y = 1, \dfrac{1}{2}$

$⇒ P ≡ \left(0, 1\right)\text{and}Q\left(\dfrac{1}{2}, \dfrac{1}{2}\right)$

$⇒ PQ = \sqrt{{\left(0 - \dfrac{1}{2}\right)}^{2} + {\left(1 - \dfrac{1}{2}\right)}^{2}}$

$⇒ PQ = \sqrt{\dfrac{1}{4} + \dfrac{1}{4}}$

$⇒ PQ = \dfrac{1}{\sqrt{2}}$