Given, x2+y2+2x=0.....(i) Centre of the given circle is C1(−1,0) and radius (r1) is 1 unit x2+y2−2y−3=0.....(ii) Centre of the circle is C2(0,1) and radius (r2) is 2
C1C2‌=√(−1)2+(1)2=√2 r1+r2‌=1+2=3‌ units ‌ C1C2‌<r1+r2
‌AC1=1‌ unit and ‌BC2=2‌ units ‌ ‌△PAC1∼△PBC2 ∴‌‌‌
PA
PB
=‌
1
2
=‌
PC1
PC2
⇒C1A is mid-point of PC2. Let the coordinate of P be (h,k) ‌‌
h+0
2
‌=−1‌ and ‌‌
k+1
2
=0 ⇒h‌=−2‌ and ‌k=−1 ∴ Coordinate of P is (−2,−1). Equation of line passes through (−2,−1) is ‌y+1=m(x+2) ⇒‌‌mx−y+2m−1=0 As PB⟂BC2, ‌‌
m×0−1+2m−1
√1+m2
‌=±2 ⇒‌‌
2(m−1)
√1+m2
‌=±2 ⇒‌‌
m−1
√1+m2
‌=±1 ⇒‌m2−2m+1‌=1+m2 ⇒‌2m‌=0 ⇒‌m‌=0 ∴ Equation of PA is y=−1 Slope of C1C2(m1)=1 tan‌θ‌=|‌
1−0
1+1×0
|=1 θ‌=45∘ ∠APM=90∘ Let m2 be the slope of line PM. Equation of line PM is ‌y+1‌=‌
1
0
(x+2) ⇒‌x‌=−2 ∴ The combined common tangent is ‌(y+1)(x+4=0 ⇒‌‌xy+2y+x+2=0