The Locus Of The Centre Of A Circle Which Touches The Circles |z – Z1| =a And |z – Z2| = B Externally (z, Z1 and Z2 are Complex Numbers) Will Be

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Question

medium

Correct option is

None of These

Let A (z₁), B(z₂) be the centres of given circles and p be the centre of the variable circle which touches given circles externally , then

|AP| = a + r and |BP| = b + r, where are is the radius of the variable circle. On subtraction, we get |AP| – |BP| = a – b

(i) right bisector of [AB] if a = b.

(ii) a hyperbola if |a – b| < | AB | = |z₂ – z₁|

(iii) an empty set if |a – b| > |AB| = |z₂ – z₁|

(iv) Set of all points on line AB except those which lie between and B if |a – b| = |AB| ≠ 0.