The circle and parabola will have common tangent atP (1, 2).
So, equation of tangent to parabola is, y×(2)=
4(x+1)
2
⇒2y=2x+2⇒y=x+1 Let equation of circle (by family of circles) is (x−x1)2+(y−y1)2+λT=0 ⇒c≡(x−1)2+(y−2)2+λ(x−y+1)=0 ∵ circles touches x -axis. ∴ y-coordinate of centre = radius ⇒c=x2+y2+(λ−2)x+(−λ−4)y+(λ+5)=0
λ+4
2
=√(
λ−2
2
)2+(
−λ−4
2
)2−(λ+5) ⇒
λ2−4λ+4
4
=λ+5⇒λ2−4λ+4=4λ+20 ⇒λ2−8λ−16=0⇒λ=4±4√2 ⇒λ=4−4√2(∵λ=4+4√2 forms bigger circle) Hence, centre of circle (2√2−2,4−2√2) and radius =4−2√2 ∴ area =π(4−2√2)2=8π(3−2√2)