Question

If $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$ represents a equation of circle, then the value of $\lambda$ when the radius of circle is minimum, is

Answer 1

Given : $|z-2-3i{|}^2+|z-4-3i{|}^2=\lambda$
Let $z=x+iy$, then

$(x-2{)}^2+(y-3{)}^2+(x-4{)}^2+(y-3{)}^2=\lambda$
$\Rightarrow 2x^2+2y^2-12x-12y=\lambda -38$
$\Rightarrow x^2+y^2-6x-6y-\frac{\lambda -38}{2}=0$

We know the equation of circle is $r=\sqrt{g^2+f^2-c}$ for the circle with eq: $x^2+y^2+2gx+2fy+c=0$

The minimum possible radius of the circle is $0$,

so, $\sqrt{9+9+\frac{\lambda -38}{2}}=0$
$\therefore \lambda =2$