Represent graphically the set of values of z for which |z−1|^2+|z−3|^2=4

Answer 1

Let z = a + b i , a, b real numbers

Then | z - 1 | = | a -1 + b i | = [ (a -1 )² + b² ]^1/2 and

| z - 3 | = | a -3 + b i | = [ (a -3 )² + b² ]^1/2

Now we are given that | z - 1 |² + | z - 3 |² = 4

{ [ (a -1 )² + b² ]^1/2 }² +{ [ (a -3 )² + b² ]^1/2}² = 4

[ (a -1 )² + b² ] + [ (a -3 )² + b² ] = 4

(a -1 )² + b² + (a -3 )² + b² = 4

2 a² - 8 a + 10 + 2 b² = 4

2 [ a² - 4 a + 4 ] + 2 + 2 b² = 4

[ a² - 4 a + 4 ] + b² = 1

[ a -1 ]² + b² =1

That is a circle centered at ( 2. 0 ) with radius 1.