Statement I : For any two non-zero complex numbers and Statement II : If are three distinct complex numbers and a, b, c are three positive real numbers such that [2024]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Consider the following two statements

Statement I : For any two non-zero complex numbers $z_{1}, z_{2},$ $(| z_{1} | + | z_{2} |)$ $| \frac{z_{1}}{| z_{1} |} + \frac{z_{2}}{| z_{2} |} |$ $\leq 2 (| z_{1} | + | z_{2} |),$ and

Statement II : If $x, y, z$ are three distinct complex numbers and a, b, c are three positive real numbers such that $\frac{a}{| y - z |} = \frac{b}{| z - x |} = \frac{c}{| x - y |},$ then $\frac{a^{2}}{y - z} + \frac{b^{2}}{z - x} + \frac{c^{2}}{x - y} = 1 .$

Between the above two statements, [2024]

1 Statement I is correct but Statement II is incorrect.

2 Both Statement I and Statement II are incorrect.

3 Both Statement I and Statement II are correct.

4 Statement I is incorrect but Statement II is correct.

Ans.
(1)

We have, $(| z_{1} | + | z_{2} |)$ $| \frac{z_{1}}{| z_{1} |} + \frac{z_{2}}{| z_{2} |} |$ $\leq (| z_{1} | + | z_{2} |) (\frac{| z_{1} |}{| z_{1} |} + \frac{| z_{2} |}{| z_{2} |})$

$\Rightarrow (| z_{1} | + | z_{2} |)$ $| \frac{z_{1}}{| z_{1} |} + \frac{z_{2}}{| z_{2} |} |$ $\leq 2 (| z_{1} | + | z_{2} |)$

Hence, statement-I is correct.

Now, let $\frac{a}{| y - z |} = \frac{b}{| z - x |} = \frac{c}{| x - y |} = λ$

$\Rightarrow a = λ | y - z |, b = λ | z - x |, c = λ | x - y |$

$\Rightarrow a^{2} = λ^{2} (y - z) (\bar{y} - \bar{z})$ [ $∵$ $| x - y |^{2} = (x - y) (\bar{x - y})$ for any complex number $x$ and $y$ ]

$b^{2} = λ^{2} (z - x) (\bar{z} - \bar{x})$

and $c^{2} = λ^{2} (x - y) (\bar{x} - \bar{y})$

Now, $\frac{a^{2}}{y - z} + \frac{b^{2}}{z - x} + \frac{c^{2}}{x - y} = λ^{2} (\bar{y} - \bar{z} + \bar{z} - \bar{x} + \bar{x} - \bar{y}) = 0$

Hence, statement-II is incorrect.

Want Us to Email you About Special Offers & Updates?