Comprehension
Read the following write up carefully answer the following questions: The complex slope of a line passing through two points represented by complex numbers \( z_{1} \) and \( z_{2} \) is defined by \( \frac{z_{2}-z_{1}}{z_{2}-z_{1}} \) and we shall denote by \( \omega \). If \( z_{0} \) is complex number and \( c \) is a real number, then \( \bar{z}_{0} z+z_{0} \bar{z}+c=0 \) represents a straight line. Its complex slope is \( -\frac{z_{0}}{\bar{z}_{0}} \).
Now consider two lines \( \alpha \bar{z}+\bar{\alpha} z+i \beta=0 \)
and \( a \bar{z}+\bar{a} z+b=0 \)
where \( \alpha, \beta \) and \( a, b \) are complex constants and let their complex slopes be denoted by \( \omega_{1} \) and \( \omega_{2} \) respectively.
Which of the following must be true?
(a) a must be purely imaginary
(b) \( \beta \) must be purely imaginary
(c) \( a \) must be real
(d) \( b \) must be imaginary
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