Equation of any circle through (0,0) and (1, 0) is

(x – 0) (x – 1) + (y – 0) (y – 0) + λ = 0

⇒ x 2 + y 2 – x + λ y = 0

If it represents C 3 , its radius = 1

⇒ 1 = (1/4) + ( λ 2 /4) ⇒ λ = ±

As the centre of C 3 , lies above the x-axis, we take λ = – and thus an equation of C 3 is x 2 + y 2 – x – y = 0

Since C 1 and C 3 intersect and are of unit radius, their common tangents are parallel to the line joining their centres (0, 0) and (1/2, /2).

So, let the equation of a common tangent be

x – y + k = 0 It will touch C 1 , if = 1 ⇒ k = ± 2

From the figure, we observe that the required tangent makes positive intercept on the y-axis and negative on the x-axis and hence its equation is – y + 2 = 0 .