Given,
O(0,0) and
A(1,0) are the centres of 2 units circles
C1 and
C3.
The equation of circle passes through
O(0,0) and
A(1,0) is
The equation of circle passes through
O(0,0) and
A(1,0) is
If it represents another unit circle
C3, its radius
=1 Here, g=−,f=,c=0 ⇒r=√+−0⇒1=+⇒λ2=4−1⇒λ=±√3 As the centre of
C3 lies above
X-axis,
f is negative, so
λ is also negative.
∴λ=−√3 Equation of circle is
x2+y2−x−√3y=0 Since,
C1 and
C3 intersect and are of unit radius, their common tangent are parallel to the line joining the centres
(0,0) and
(1/2,√3/2).
Let the equation of a common tangent be
√3x−y+k=0 It will touch
C1 if
=1k=±2 ∴ The required positive tangent make positive intercept on
Y-axis and negative on
X-axis and hence, the required equation is
√3x−y+2=0