If |z - 2|/|z - 3| = 2 represents a circle, then its radius is eq…

MCQ

If $|z - 2|/|z - 3| = 2$ represents a circle, then its radius is equal to

A
$1$
B
$1/3$
C
$3/4$
✓
$2/3$

✓

Answer

Correct option: D.

$2/3$

d
(d)Given, $\frac{{|z - 2|}}{{|z - 3|}} = 2$
==> $\sqrt {{{(x - 2)}^2} + {y^2}} = 2\sqrt {{{(x - 3)}^2} + {y^2}} $
==> ${(x - 2)^2} + {y^2} = 4[{(x - 3)^2} + {y^2}]$
==> ${x^2} + {y^2} + 4 - 4x = 4{x^2} + 4{y^2} + 36 - 24x$
==> $3{x^2} + 3{y^2} - 20x + 32 = 0$
or ${x^2} + {y^2} - \frac{{20}}{3}x + \frac{{32}}{3} = 0$ .....$(i)$
We know that, standard equation of circle,
${x^2} + {y^2} + 2gx + 2fy + c = 0$ .....$(ii)$
Comparison of $(i)$ from $(ii)$
==> $2g = - \frac{{20}}{3}\, \Rightarrow g = - \frac{{10}}{3},f = 0,c = \frac{{32}}{3}$
Hence, Radius =$\sqrt {{g^2} + {f^2} - c} $$ = \sqrt {\frac{{100}}{9} - \frac{{32}}{3}} = \sqrt {\frac{4}{9}} = \frac{2}{3}$

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