Question

The radius of the largest circle, which passes through the focus of the parabola $y^2=4(x+y)$ and also contained in it, is

Answer 1

$y^2=4(x+y)$
$\Rightarrow (y-2{)}^2=4(x+1)...(1)$
Its focus is $(0,2).$
Let radius of the circle be $r.$
Then, equation of circle is
$(x-r{)}^2+(y-2{)}^2=r^2...\left(2\right)$
Solving eqns. $\left(1\right)$ and $\left(2\right),$ we get
$(x-r{)}^2+4(x+1)=r^2$
$\Rightarrow x^2+(4-2r)x+4=0$
$\Rightarrow (4-2r{)}^2-16=0[\because D=0]$
$\Rightarrow 4-2r=\pm 4$
$\Rightarrow r=4$