Question

if z1 = 4-3i and z2 = -12+5i show that |z1+z2| greater than |z1| + |z2|​

Answer 1

= zi = 4-3i , z2 = 12 + 5i

= z1/z2= 4-3i / 12+5i = 4- 3i / 12+5i × 12-5i / 12-5i

= (4- 3i) (12-5i ) / (12)²- (5i)²

= 48-20i - 36i + 15i²/ 144+25 ...( i2 = -1)

....( a+b) (a-b) = a²-b²

= 33-56i / 169 ....( z= x+iy , 121= Root x² + y²)

=|z1 /z2| =1/ 169 Root 33² +56 ²

=1/169 Root 1089 +3136

= 1/169 Root 4225

=65/169 =5/13

=|Z1| /|Z2| = |4-3i| / |12+5i | = Root 4²+3² / 12²+5²

= Root 25/ Root 169

=5/ 13

= so | Z| / |Z2 | = |Z1|/ |Z2| proved