If z1 and z2 be two distinct complex numbers satisfying z12-z22=z12+z22-2z1z2 and if arg z1-arg z2=aπb, then find the least possible value of a+ba,b∈integers.

z12-z22=z12+z22-2z1z2 z12-z22=z1-z22 ⇒z1+z2=z1-z2 z1⊥z2 ⇒argz1z2=2nπ±π2⇒-π2,π2 Minimum value of a+b=1 where a=-1 and b=2.

If z1 and z2 be two distinct complex numbers satisfying z12-z22=z12+z22-2z1z2 and if arg z1-arg z2=aπb, then find the least possible value of a+ba,b∈integers.

z12-z22=z12+z22-2z1z2 z12-z22=z1-z22 ⇒z1+z2=z1-z2 z1⊥z2 ⇒argz1z2=2nπ±π2⇒-π2,π2 Minimum value of a+b=1 where a=-1 and b=2.

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If $z_{1}$ and $z_{2}$ be two distinct complex numbers satisfying $|{z_{1}}^{2} - {z_{2}}^{2}| = |{z_{1}}^{2} + z_{2}^{2} - 2 z_{1} z_{2}|$ and if $(a r g z_{1} - a r g z_{2}) = \frac{a π}{b}$ , then find the least possible value of $|a + b| (a, b \in i n t e g e r s)$ .

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Detailed Solution

$|{z_{1}}^{2} - {z_{2}}^{2}| = |{z_{1}}^{2} + z_{2}^{2} - 2 z_{1} z_{2}| |{z_{1}}^{2} - {z_{2}}^{2}| = |{(z_{1} - z_{2})}^{2}| \Rightarrow |z_{1} + z_{2}| = |z_{1} - z_{2}| z_{1} ⊥ z_{2} \Rightarrow a r g (\frac{z_{1}}{z_{2}}) = 2 n π \pm \frac{π}{2} \Rightarrow \frac{- π}{2}, \frac{π}{2}$