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A straight line is passing through the points represented by
the complex numbers a $+\mathrm{ib}$ and $\frac{1}{-\mathrm{a}+\mathrm{ib}}$, where $(\mathrm{a}, \mathrm{b}) \neq$ $(0,0)$
Which one of the following is correct?
Options:
the complex numbers a $+\mathrm{ib}$ and $\frac{1}{-\mathrm{a}+\mathrm{ib}}$, where $(\mathrm{a}, \mathrm{b}) \neq$ $(0,0)$
Which one of the following is correct?
Solution:
1793 Upvotes
Verified Answer
The correct answer is:
It passes through the origin
Complex numbers are : $(\mathrm{a}+\mathrm{ib})$ and $\frac{1}{-\mathrm{a}+\mathrm{ib}}$
Second number is rationalized as,
$\frac{-a-i b}{(-a+i b)(-a-i b)}=\frac{-a-i b}{a^{2}+b^{2}}$
These two complex numbers represent two points,
$(\mathrm{a}, \mathrm{b})$ and $\left(\frac{-\mathrm{a}}{\mathrm{a}^{2}+\mathrm{b}^{2}}, \frac{-\mathrm{b}}{\mathrm{a}^{2}+\mathrm{b}^{2}}\right)$
Eq. of line passing through these point is
$(\mathrm{y}-\mathrm{b})=\frac{-\frac{\mathrm{b}}{\mathrm{a}^{2}+\mathrm{b}^{2}}-\mathrm{b}}{\frac{-\mathrm{a}}{\mathrm{a}^{2}+\mathrm{b}^{2}}-\mathrm{a}}(\mathrm{x}-\mathrm{a})$
$\Rightarrow \mathrm{y}-\mathrm{b}=\frac{\mathrm{b}}{\mathrm{a}}(\mathrm{x}-\mathrm{a}) \Rightarrow \mathrm{ay}-\mathrm{ab}=\mathrm{bx}-\mathrm{ab}$
$\Rightarrow \mathrm{y}=\frac{\mathrm{b}}{\mathrm{a}} \mathrm{x}$
So, line passes through the origin.
Second number is rationalized as,
$\frac{-a-i b}{(-a+i b)(-a-i b)}=\frac{-a-i b}{a^{2}+b^{2}}$
These two complex numbers represent two points,
$(\mathrm{a}, \mathrm{b})$ and $\left(\frac{-\mathrm{a}}{\mathrm{a}^{2}+\mathrm{b}^{2}}, \frac{-\mathrm{b}}{\mathrm{a}^{2}+\mathrm{b}^{2}}\right)$
Eq. of line passing through these point is
$(\mathrm{y}-\mathrm{b})=\frac{-\frac{\mathrm{b}}{\mathrm{a}^{2}+\mathrm{b}^{2}}-\mathrm{b}}{\frac{-\mathrm{a}}{\mathrm{a}^{2}+\mathrm{b}^{2}}-\mathrm{a}}(\mathrm{x}-\mathrm{a})$
$\Rightarrow \mathrm{y}-\mathrm{b}=\frac{\mathrm{b}}{\mathrm{a}}(\mathrm{x}-\mathrm{a}) \Rightarrow \mathrm{ay}-\mathrm{ab}=\mathrm{bx}-\mathrm{ab}$
$\Rightarrow \mathrm{y}=\frac{\mathrm{b}}{\mathrm{a}} \mathrm{x}$
So, line passes through the origin.
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