Question

Assertion :The locus of the centre of a circle which touches the circles $|z-z_1|=a$ & $|z-z_2|=b$ externally. ($z,z_1,z_2$ are complex numbers) will be hyperbola. Reason: $|z-z_1|-|z-z_2|<|z_2-z_1|$ $\Rightarrow z$ lies on the hyperbola.

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

Answer 1

The correct option is D (A) is false (R) is true.

Let $A\left(z_1\right)$ & $B\left(z_2\right)$ be the centre of given circles

and $P$ be the centre of the variable circle which touches the given circle externally then $AP=a+r,BP=b+r$
Where ${}^{\prime }{r}^{\prime }$ is the radius of variable circle, on subtracting $|AP|-|BP|=|a-b|$
$\Rightarrow ||AP|-|BP||=|a-b|$ is a constant
Hence locus of $P$ is
(i) An empty set if $|a-b|>|AB|=|z_2-z_1|$
(ii) A Right bisector of $AB$ if $a=b$
(iii) A hyperbola if $|a-b|<|AB|=|z_2-z_1|$
(iv) Set of all points on the line $AB$ except those which lies between $A$ & $B$ if $|a-b|=|AB|\ne 0$
$\therefore$ Assertion (A) is false & Reason (R) is true.