O(0,0) and A(1,0) are centres of two unit circles C1 and C2 respectively. C3 is also a unit circle having its centre above X-axis and passing through O and A. The equation of the common tangent to C1 and C3 which does not intersect the circle C2 is
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
‌(x−0)(x−1)+(y−0)(y−0)+λ|
x
y
1
0
0
1
1
0
1
|=0 ‌⇒x2−x+y2+λy=0⇒x2+y2−x+λy=0
If it represents another unit circle C3, its radius =1 ‌ Here, ‌g=−‌
1
2
,f=‌
λ
2
,c=0 ⇒r=√‌
1
4
+‌
λ2
4
−0⇒1=‌
1
4
+‌
λ2
4
⇒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
|k|
√(√3)2+(−1)2
=1 ‌k=±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