Complex Numbers And Quadratic Equations question 331
Question: If $ |z-2|/|z-3|=2 $ represents a circle, then its radius is equal to [Kurukshetra CEE 1998]
Options:
A) 1
B) $ 1/3 $
C) $ 3/4 $
D) $ 2/3 $
Show Answer
Answer:
Correct Answer: D
Solution:
Given, $ \frac{|z-2|}{|z-3|}=2 $
Þ $ \sqrt{{{(x-2)}^{2}}+y^{2}}=2\sqrt{{{(x-3)}^{2}}+y^{2}} $
Þ $ {{(x-2)}^{2}}+y^{2}=4[{{(x-3)}^{2}}+y^{2}] $
Þ $ x^{2}+y^{2}+4-4x=4x^{2}+4y^{2}+36-24x $
Þ $ 3x^{2}+3y^{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)
Þ $ 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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