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Question

Assertion :The locus of the centre of a circle which touches the circles |z−z1|=a & |z−z2|=b externally. (z,z1,z2 are complex numbers) will be hyperbola. Reason: |z−z1|−|z−z2|<|z2−z1| ⇒z lies on the hyperbola.

A
Both (A) & (R) are individually true & (R) is correct explanation of (A).
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B
Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
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C
(A) is true (R) is false.
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D
(A) is false (R) is true.
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Solution

The correct option is D (A) is false (R) is true.Let A(z1) & B(z2) be the centre of given circles and P be the centre of the variable circle which touches the given circle externally then AP=a+r,BP=b+rWhere ′r′ is the radius of variable circle, on subtracting |AP|−|BP|=|a−b|⇒||AP|−|BP||=|a−b| is a constantHence locus of P is(i) An empty set if |a−b|>|AB|=|z2−z1|(ii) A Right bisector of AB if a=b(iii) A hyperbola if |a−b|<|AB|=|z2−z1|(iv) Set of all points on the line AB except those which lies between A & B if |a−b|=|AB|≠0∴ Assertion (A) is false & Reason (R) is true.

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