Question

The line $x=y$ touches a circle at the point $(1,1)$. If the circle also passes through the point $(1,-3)$ then its radius is:

• A. $3\sqrt{2}$
• B. $3$
• C. $2\sqrt{2}$
• D. $2$

Answer 1

Answer: (A)

$3\sqrt{2}$

Equation of circle can be written as
$(x-1{)}^2+(y-1{)}^2+\lambda (x-y)=0$
If passes through $(1,3)$
$16+\lambda \left(4\right)=0\Rightarrow \lambda =-4$
So $(x-1{)}^2+(y-1{)}^2-4(x-y)=0$
$\Rightarrow x^2+y^2-6x+2y+2=0$
$\Rightarrow r=2\sqrt{2}$