Two circles of radii 'a' and 'b' touching each other externally, are inscribed in the area bounded by y=√1−x2 and the x-axis. If b=12, then a is equal to
Given area x2+y2=1 and x-axis
i.e. upper semicircle
According to the diagram
B1D=B2D=1
C1D=1−a,C2D=1−b=12
A1C1=a,A2C2=b=12
∠C1A1D=∠C2A2D=90∘
By Pythagoras theorem
A1D=√(C1D)2–(A1C1)2
=√(1–a)2–a2=√1−2a
A2D=√14−14=0
C1=(A1D,a)=(√1–2a,a)
C2=(A2D,b)=(0,12)
Since the circles touch each other
C1C2=r1+r2
C1C22=(a+12)2
⟹(1–2a)+(a−12)2=(a+12)2
⟹1–2a=2a
⟹a=14