I am unable to solve this easy question of complex number.Please help me.
Let a,b,c be distinct complex number such that:
$$\frac{a}{1-b}=\frac{b}{1-c}=\frac{c}{1-a}=k$$
then find k?
Let a,b,c be distinct complex number such that (a/1-b)=(b/1-c)=(c/1-a)=k then find k?
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complex-numbers
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2How come you can't solve it, since you call it *easy*? – 2017-01-17
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0Use the first equation to eliminate $a$, (find it in terms of $b,k,c$) then use the second equation to eliminate $b$ (find it in terms of $c,k$). Then you will get an equation you can solve. – 2017-01-17
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0See http://mathworld.wolfram.com/CramersRule.html – 2017-01-17
1 Answers
1
We know that:
$$\text{k}=\frac{\text{a}}{1-\text{b}}\space\Longleftrightarrow\space\text{a}=\text{k}\left(1-\text{b}\right)$$
When $\text{b}\ne-1$
We also know that:
$$\text{k}=\frac{\text{b}}{1-\text{c}}\space\Longleftrightarrow\space\text{b}=\text{k}\left(1-\text{c}\right)$$
When $\text{c}\ne-1$
And:
$$\text{k}=\frac{\text{c}}{1-\text{a}}\space\Longleftrightarrow\space\text{a}=1-\frac{\text{c}}{\text{k}}$$
When $\text{k}\ne0\space\wedge\space\text{c}\ne0$
So, we can set:
$$1-\frac{\text{c}}{\text{k}}=\text{k}\left(1-\text{k}\left(1-\text{c}\right)\right)\space\Longleftrightarrow\space\text{k}=\frac{1}{2}\pm\frac{i\sqrt{3}}{2}$$