Complex Numbers And Quadratic Equations question 140
Question: If $ z_1=10+6i,z_2=4+6i $ and $ z $ is a complex number such that $ amp( \frac{z-z_1}{z-z_2} )=\frac{\pi }{4}, $ then the value of $ |z-7-9i| $ is equal to [IIT 1990]
Options:
A) $ \sqrt{2} $
B) $ 2\sqrt{2} $
C) $ 3\sqrt{2} $
D) $ 2\sqrt{3} $
Show Answer
Answer:
Correct Answer: C
Solution:
Given numbers are $ z_1=10+6i,z_2=4+6i $ and $ z=x+iy $ \ $ amp( \frac{z-z_1}{z-z_2} )=\frac{\pi }{4} $
Þ $ amp[ \frac{(x-10)+i(y-6)}{(x-4)+i(y-6)} ]=\frac{\pi }{4} $
Þ $ \frac{(x-4)(y-6)-(y-6)(x-10)}{(x-4)(x-10)+{{(y-6)}^{2}}}=1 $
Þ $ 12y-y^{2}-72+6y=x^{2}-14x+40 $ …..(i) Now $ |z-7-9i|=|(x-7)+i(y-9)| $
Þ $ \sqrt{{{(x-7)}^{2}}+{{(y-9)}^{2}}} $ ….(ii) From (i), $ (x^{2}-14x+49)+(y^{2}-18y+81)=18 $
Þ $ {{(x-7)}^{2}}+{{(y-9)}^{2}}=18 $ or $ {{[{{(x-7)}^{2}}+{{(y-9)}^{2}}]}^{1/2}}={{[18]}^{1/2}}=3\sqrt{2} $ \ $ |(x-7)+i(y-9)|=3\sqrt{2} $ or $ |z-7-9i|=3\sqrt{2} $ .
BETA
