The circle passing through $\left(t,1\right),\left(1,t\right)$ and $\left(t,t\right)$ for all values of $t$ also passes through ;

The general equation of circle is

$(x-h)^{2}+(y-k)^{2} = r^{2},$ where h and k are the

coordinates

of center of circle.

if It passes through $(1,t)$ so $(1-h)^{2}+(t-k)^{2} = r^{2}$ __ (1)

if It passes through $(t,t)$ so $(t-h)^{2}+(t-k)^{2} = r^{2}$ __ (2)

if It passes through $(t,1)$ so $(t-h)^{2}+(1-k)^{2} = r^{2}$ __ (3)

on solving eq (1) & (2)

$1-h = t-h \Rightarrow t = 1$

It t = 1, then the circle passes through (1,1).