Question

For what value of $\lambda$, does the line $x+y=\lambda$ touches the circle $x^2+y^2-2x-2y=0$

Answer 1

$x+y=\lambda$

touches

$x^2+y^2-2x-2y=0$

$(x-1{)}^2+(y-1{)}^2=2$

$(1,1)$ centre and radius $\sqrt{2}$

condition for line to touch circle is

$r=\frac{C}{\sqrt{m^2+1}}$

$\sqrt{2}=\frac{\lambda }{\sqrt{2}}$

$\lambda =2$