Circles Part 1

Given, S1⇒x2+y2=9
S2⇒(x−2)2+y2=1
∴‌‌C1≡(0,0) and r1=3
C2≡(2,0),r2=1
Let centre of variable circle be P(h,k) and radius be r.
Let centre of variable circle be P(h,k) and radius be r.
From the given data, it is clear that given circles touch each other internally. Hence, variable circle touches S1 internally and S2 externally as shown in figure below.Clearly, PC1=3−r
and PC2=1+r
and C1C2=2
∴‌‌PC1+PC2=4
which is greater than C1C2.
Hence, locus of P is an ellipse whose foci are C1 and C2 and major axis is 2a=4.
and ‌‌2ae=C1C2=2
⇒‌‌e=‌

1

2

⇒‌‌b2=4(1−‌

1

4

)=3
Centre is mid-point of C1(0,0) and C2(2,0).
∴ Centre of ellipse is (1,0).
‌

(x−1)2

4

+‌

y2

3

=1 is the equation of locus of centre of variable circle which is satisfied by (2,±‌

3

2

).