Question-19592

Question Number 19592 by ajfour last updated on 13/Aug/17

Commented by Tinkutara last updated on 13/Aug/17

z_{F} = \frac{{a b}^{2}}{a^{2} + b^{2}} + i \frac{a^{2} b}{a^{2} + b^{2}}

Commented by ajfour last updated on 13/Aug/17

If we seek to find equation of

line shown above, z = z_{F} + i λ z_{F}

and \bar{z} = {\bar{z}}_{F} - i λ {\bar{z}}_{F}

so \frac{z - z_{F}}{\bar{z} - {\bar{z}}_{F}} = - \frac{z_{F}}{{\bar{z}}_{F}}

\Rightarrow {\bar{z}}_{F} z - ∣ z_{F} ∣^{2} = - z_{F} \bar{z} + ∣ z_{F} ∣^{2}

\Rightarrow {\bar{z}}_{F} z + z_{F} \bar{z} - 2 ∣ z_{F} ∣^{2} = 0

if equation of line be

\bar{α} z + α \bar{z} + 2 c = 0

then this means α = z_{F} and c = - ∣ z_{F} ∣^{2} .

complex slope = - \frac{α}{\bar{α}} = - \frac{z_{F}}{{\bar{z}}_{F}} .

= - (\frac{b + ia}{b - ia}) \times \frac{i}{i} = \frac{a - ib}{a + ib} .

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