SATHEE: Complex Numbers 3 Question 9

Complex Numbers 3 Question 9

9.

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $\arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then

(2010)

(a) $| z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

(b) $\arg (z - z_{1}) = \arg (z - z_{2})$

(c) $| \begin{array}{cc} z - z_{1} & \bar{z} - {\bar{z}}_{1} \\ z_{2} - z_{1} & {\bar{z}}_{2} - {\bar{z}}_{1} \end{array} | = 0$

(d) $\arg (z - z_{1}) = \arg (z_{2} - z_{1})$

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Answer:

Correct Answer: 9. $(a, c, d)$

Solution:

Given, $z = \frac{(1 - t) z_{1} + t z_{2}}{(1 - t) + t}$

Clearly, $z$ divides $z_{1}$ and $z_{2}$ in the ratio of $t : (1 - t)$ , $0 < t < 1$

$\Rightarrow A P + B P = A B$ i.e. $| z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

$\Rightarrow$ Option (a) is true.

$and \begin{aligned} \arg (z - z_{1}) & = \arg (z_{2} - z) \\ = \arg (z_{2} - z_{1}) \end{aligned}$

$\Rightarrow$ Option (b) is false and option (d) is true.

Also, $\arg (z - z_{1}) = \arg (z_{2} - z_{1})$

$\Rightarrow \arg \frac{z - z_{1}}{z_{2} - z_{1}} = 0$

$∴ \frac{z - z_{1}}{z_{2} - z_{1}}$ is purely real.

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{\bar{z} - {\bar{z}}_{1}}{{\bar{z}}_{2} - {\bar{z}}_{1}}$

or $| \begin{array}{cc} z - z_{1} & \bar{z} - {\bar{z}}_{1} \\ z_{2} - z_{1} & {\bar{z}}_{2} - {\bar{z}}_{1} \end{array} | = 0$

Option (c) is correct.

Complex Numbers 3 Question 9

9.

Answer:

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Complex Numbers 3 Question 9

9.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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