Complex Numbers And Quadratic Equations question 382
Question: If $ |z-2-3i|+|z+2-6i|=4 $ , where $ i=\sqrt{-1} $ , then locus of $ P(z) $ is [DCE 2005]
Options:
A) An ellipse
B) $ \varphi $
C) Line segment joining of point $ 2+3i $ and $ -2+6i $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
$ |z-z_1|+|z-z_2|=2a $ when $ |z_1-z_2|\le 2a $ , then it is an ellipse $ z_1=2+3i $ and $ z_2=-2+6i $ $ z_1-z_2=(2+3i)-(-2+6i)=4-3i $ $ |z_1-z_2|=|4-3i| $ = $ \sqrt{4^{2}+{{(-3)}^{2}}}=5 $ But $ 5<4 $ is false, because in any triangle sum of two sides is not smaller than third side.
$ \therefore $ $ P(z) $ is not represent locus of any point.
BETA
