Question

The circle passing through $(t,1)$, $(1,t)$ and $(t,t)$
for all values of $t$ passes through .........

• A. $(1,1)$
• B. $(-1,-1)$
• C. $(1,-1)$
• D. $(-1,1)$

Answer 1

The correct option is A $(1,1)$

Let $x^2+y^2+2gn+2fy+c=0e{q}^n$ of circle.
So, (t,1) $\Rightarrow t^2+2gt+2f+1+c=0----\left(1\right)$
(1,t) $\Rightarrow t^2+2ft+2g+1+c=0----\left(2\right)$
(t,t) $\Rightarrow 2t^2+2t(g+f)+c=0----\left(3\right)$
from (1) & (2)
$(g-f)t+(f-g)=0$
$t=t_1$
So, P $(t,t)$ is $(1,1).$
So, circles passing through $(1,t),(t,1)$ and $(t,t)$ always passes through $(1,1).$