SOLUTION: COMPLEX NUMBER If {{{z = a+bi}}} and {{{abs((z-i)/(z+2+i))=3}}}, show that {{{8a^2 + 8b^2 + 36a +20b +44 =0}}} Thanks.

Algebra ->  Equations -> SOLUTION: COMPLEX NUMBER If {{{z = a+bi}}} and {{{abs((z-i)/(z+2+i))=3}}}, show that {{{8a^2 + 8b^2 + 36a +20b +44 =0}}} Thanks.      Log On


   

Question 976172: COMPLEX NUMBER
If z+=+a%2Bbi and abs%28%28z-i%29%2F%28z%2B2%2Bi%29%29=3, show that 8a%5E2+%2B+8b%5E2+%2B+36a+%2B20b+%2B44+=0
Thanks.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
COMPLEX NUMBER
If z+=+a%2Bbi and abs%28%28z-i%29%2F%28z%2B2%2Bi%29%29=3, show that 8a%5E2+%2B+8b%5E2+%2B+36a+%2B20b+%2B44+=0 
abs%28%28z-i%29%2F%28z%2B2%2Bi%29%29%22%22=%22%223

abs%28+%28%28a%2Bbi%29-i%29%2F%28%28a%2Bbi%29%2B2%2Bi%29%29%22%22=%22%223

abs%28%28a%2Bbi-i%29%2F%28a%2Bbi%2B2%2Bi%29%29%22%22=%22%223

abs%28%28a%2B%28bi-i%29%29%2F%28a%2B2%2B%28bi%2Bi%29%29%29%22%22=%22%223

abs%28%28a%2B%28b-1%29i%29%2F%28%28a%2B2%29%2B%28b%2B1%29i%29%29%22%22=%22%223

sqrt%28a%5E2%2B%28b-1%29%5E2%29%2Fsqrt%28%28a%2B2%29%5E2%2B%28b%2B1%29%5E2%29%22%22=%22%223

Square both sides:

%28a%5E2%2B%28b-1%29%5E2%29%2F%28%28a%2B2%29%5E2%2B%28b%2B1%29%5E2%29%22%22=%22%229

a%5E2%2B%28b-1%29%5E2%22%22=%22%229%28%28a%2B2%29%5E2%2B%28b%2B1%29%5E2%29

a%5E2%2B%28b-1%29%5E2%22%22=%22%229%28a%2B2%29%5E2%2B9%28b%2B1%29%5E2%29

a%5E2-9%28a%2B2%29%5E2%22%22=%22%229%28b%2B1%29%5E2-%28b-1%29%5E2

Both sides are differences of squares, so factor:

%28a-3%28a%2B2%29%5E%22%22%29%28a%2B3%28a%2B2%29%5E%22%22%29%22%22=%22%22

%28a-3%28a%2B2%29%29%28a%2B3%28a%2B2%29%29%22%22=%22%22%283%28b%2B1%29-%28b-1%29%29%283%28b%2B1%29%2B%28b-1%29%29

%28a-3a-6%29%28a%2B3a%2B6%29%22%22=%22%22%283b%2B3-b%2B1%29%283b%2B3%2Bb-1%29

%28-2a-6%29%284a%2B6%29%22%22=%22%22%282b%2B4%29%284b%2B2%29

%28-2%28a%2B3%29%5E%22%22%29%282%282a%2B3%29%5E%22%22%29%22%22=%22%22%282%28b%2B2%29%5E%22%22%29%282%282b%2B1%29%5E%22%22%29

-4%28a%2B3%29%282a%2B3%29%22%22=%22%224%28b%2B2%29%282b%2B1%29

-%28a%2B3%29%282a%2B3%29%22%22=%22%22%28b%2B2%29%282b%2B1%29

-%282a%5E2%2B9a%2B9%29%22%22=%22%222b%5E2%2B5b%2B2

-2a%5E2-9a-9%22%22=%22%222b%5E2%2B5b%2B2

-2a%5E2-9a-2b%5E2-5b-11%22%22=%22%220

2a%5E2%2B9a%2B2b%5E2%2B5b%2B11%22%22=%22%220

If you swap 2nd and 3rd terms and multiply through by 4, you get 

8a%5E2+%2B+8b%5E2+%2B+36a+%2B20b+%2B44%22%22=%22%220

Edwin