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if |z1+z2|=|z1|+|z2|, then prove that a ...

if `|z_1+z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)` if `|z_1-z_2|=|z_1|+|z_2|,` then prove that `a r g(z_1)=a r g(z_2)=pi`

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(a) ` |z_1 - z _2 | = | z_1 |- |z_2 | `

` rArr AB = OA - OB ` as shown in figure
`rArr ` Point ` A(z_1 ) , O ( 0 ), B(z_2) ` are collinear
`rArr arg(z_1) = arg (z_2) = theta `
(b) ` |z_1 + z _ 2 | = |z_1| - |z_2|`
` rArr ` Points ` A(z_1), O(0), B(-z_2) ` are collinear as shown in figurr

` rArr ` Points ` A(z_1 ) , O(0) , C(z_2)` are collinear
If ` arg(z_1) = theta`, then ` arg(z_2) = theta - pi `
` rArr arg (z_1) - arg (z_2) = pi `
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