SATHEE: Complex Numbers 4 Question 15

Complex Numbers 4 Question 15

15.

Complex numbers $z_{1}, z_{2}, z_{3}$ are the vertices $A, B, C$ respectively of an isosceles right angled triangle with right angle at $C$ . Show that

${(z_{1} - z_{2})}^{2} = 2 (z_{1} - z_{3}) (z_{3} - z_{2})$ .

(1986, $2 \frac{1}{2}$ M)

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Solution:

Since, triangle is a right angled isosceles triangle.

$∴$ Rotating $z_{2}$ about $z_{3}$ in anti-clockwise direction through an angle of $π / 2$ , we get

where, $| z_{2} - z_{3} | = | z_{1} - z_{3} |$

$\Rightarrow (z_{2} - z_{3}) = i (z_{1} - z_{3})$

On squaring both sides, we get

$\begin{aligned} {(z_{2} - z_{3})}^{2} = - {(z_{1} - z_{3})}^{2} \\ \Rightarrow z_{2}^{2} + z_{3}^{2} - 2 z_{2} z_{3} = - z_{1}^{2} - z_{3}^{2} + 2 z_{1} z_{3} \\ \Rightarrow z_{1}^{2} + z_{2}^{2} - 2 z_{1} z_{2} = 2 z_{1} z_{3} + 2 z_{2} z_{3} - 2 z_{3}^{2} - 2 z_{1} z_{2} \end{aligned}$ $\begin{array}{ll} \Rightarrow & {(z_{1} - z_{2})}^{2} = 2 (z_{1} z_{3} - z_{3}^{2}) + (z_{2} z_{3} - z_{1} z_{2}) \\ \Rightarrow & {(z_{1} - z_{2})}^{2} = 2 (z_{1} - z_{3}) (z_{3} - z_{2}) \end{array}$

Complex Numbers 4 Question 15

15.

Solution:

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Complex Numbers 4 Question 15

15.

Solution:

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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