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(i) If |z(1) +z(2)| = |z(1)|+|z(2)|, the...

(i) If `|z_(1) +z_(2)| = |z_(1)|+|z_(2)|`, then prove that `arg(z_(1)) = arg(z_(2))`
(ii) If `|z_(1) -z_(2)| = |z_(1)| +|z_(2)|`,then prove that `arg(z_(1)) - arg(z_(2)) = pi`

Text Solution

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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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