Express the following polar no s in polar form 1+2i/ 1-3i 1-3i/ 1+2i - Maths - Complex Numbers and Quadratic Equations

Question

Express the following polar no s in polar form

1+2i/ 1-3i
1-3i/ 1+2i

Santosh Kumar · Accepted Answer

$\frac{1 + 2 i}{1 - 3 i} = \frac{1 + 2 i}{1 - 3 i} \times \frac{1 + 3 i}{1 + 3 i} = \frac{(1 + 2 i) (1 + 3 i)}{1 + 9} = \frac{1 + 3 i + 2 i + 6 i^{2}}{10} = \frac{5 i + 1 - 6}{10} = \frac{5 i - 5}{10} = \frac{- 1}{2} + \frac{i}{2} modulus, r = | z | = \sqrt{(\frac{- 1}{2})^{2} + (\frac{1}{2})^{2}} = \frac{1}{\sqrt{2}}$

let α be the acute angle given by,

$\tan α = | \frac{Im(z)}{Re(z)} | = | \frac{(\frac{1}{2})}{(\frac{- 1}{2})} | = 1 \Rightarrow α = \frac{π}{4}$

Since, the point $(\frac{- 1}{2}, \frac{1}{2})$ lies in the 2nd quadrant.

$∴ θ = \arg (z) = π - α = π - \frac{π}{4} = \frac{3 π}{4}$

Hence, required polar form,

$z = \frac{1}{\sqrt{2}} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})$

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Express the following polar no.s in polar form 1+2i/ 1-3i 1-3i/ 1+2i

Express the following polar no.s in polar form

1+2i/ 1-3i

1-3i/ 1+2i