Find the equation in the complex variable of the family of all circles that cut the circles |Z| = i and |Z –3| = 4 orthogonally.
Text Solution
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Sol. Let |Z – α | = r … (i)
be the circle which cuts both the circles |Z| = 1… (ii)
and |Z –3| = 4 … (iii)
orthogonally.
∴ sum of the squares of the radii (i) and (ii) = the squares of the distance between their centers.
Hence r 2 + 1 = | α –0| 2 = α 
Similarly, r 2 + 16 = | α –3| 2 + α 
–6 Re ( α ) + 9
∴ subtracting 15 = –6 Re ( α ) + 9 ⇒ Re ( α ) = –1
∴ α = –1 + ib
∴ r 2 + 1 = 1 + b 2 ∴ the required family of circles is given by
|Z – (–1 + ib)} = |b|, where b ∈ R.
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