Question

Assertion :Two lines $a\overline{z}+\overline{a}z+b=0,a_1\overline{z}+\overline{a_1}z+b_1=0$ (where $a,a_1\in C,a,a_1\ne 0$ and $b,b_1\in R$) are parallel if and only if $\frac{a}{a_1}$ is real. Reason: Two lines $a\overline{z}+\overline{a}z+b=0,a_1\overline{z}+\overline{a_1}z+b_1=0$ (where $a,a_1\in C,a,a_1\ne 0$ and $b,b_1\in R$) are perpendicular if and only if $\frac{a}{a_1}$ is purely imaginary.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given two lines, $a\overline{z}+\overline{a}z+b=0$

$a_1\overline{z}+\overline{a_1}z+b_1=0$

Two linear equations will represent parallel lines if the relation of coefficient is $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$

$\therefore$ From the given two lines, $\frac{a}{a_1}$ is real $\therefore$ the lines are parallel

In the reason, if the lines $a\overline{z}+\overline{a}z+b=0$ and $a_1\overline{z}+\overline{a_1}z+b_1=0$ are perpendicular then the relation between $a_1,a,b_1$ and $b$ should be known.

Thus, $\frac{a}{a_1}$ represent imaginary.

$\therefore$ Both assertion and reason are correct but reason is not the correct explanation of assertion.