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

Explanation

We have parabola
`y^(2)=4(x+y)or(y-2)^(2)=4(x+1)` (1)
Vertex is (-1,2) and focus is (0,2).
For the largest circle passing through (0,2) contained in the parabola, circle must be symmetrical about the axis of the parabola. So, centre lies on the line y=2.
Let radius of the circle be r.
Then equation of circle is
`(x-r)^(2)+(y-2)^(2)=r^(2)`

Solving (1) and (2), we get
`(x-r)^(2)+4(x-1)=r^(2)`
`rArr" "x^(2)+(4-2r)x+4=0`
Since largest circle toches parabola, abscissa of points A and B must be same.
So, above equation has equal roots.
`:." "(4-2r)^(2)-16=0`
`rArr" "4-2r=pm4`
`rArr" "r=4`