Question

Assertion :If $\frac{5z_2}{11z_1}$ is purely imaginary then $\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|=1$ Reason: $|z|=|\overline{z}|$ i.e., $|a+ib|=|a-ib|,i=\sqrt{-1}$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Given, $\frac{5z_2}{11z_1}$ is purely imaginary.
$\Rightarrow \frac{5z_2}{11z_1}=ki$
$\Rightarrow \frac{z_1}{z_2}=\frac{5}{11ki}$
$\Rightarrow \frac{2z_1}{3z_2}=\frac{10}{33ki}$
applying componendo and dividendo gives
$\frac{2z_1+3z_2}{2z_1-3z_2}=\frac{10+i33k}{10-i33k}$
taking mod on both sides
$\therefore \left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|=\left|\frac{10+i33k}{10-i33k}\right|=1$ Since,$|z|=|\overline{z}|$
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
Hence, option A.