Question

The circle passing through three distinct points $(1,t),(t,1)$ and $(t,t)$ for all values of $t$, also passes through the point:

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

Answer 1

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

All three distinct points $A(1,t),B(t,1)$ and $C(t,t)$ are forming a right angled isosceles triangle.
If the circle passes through these points then the center of the circle will be mid point of $AB.$
Now, image of $C$ in line $AB$ will also lie on the circle.
Hence, image of $C$ in $AB$ will be $(1,1).$
Hence, circle will always pass through $(1,1).$