mathematics-class-11-unit-05-chapter-15-complex-numbers-l-15-15-dqdckzwhm7g

<h3 id="successproblemsuccess"><Success>Problem</Success></h3>
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<p>Let $T$ be a triangle with vertices $z_1, z_4, z_3$ in $\mathbb{C}$</p>
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<p>If $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|$ and $z_1+z_2+z_3=0$. then show that $T$ is equilateral triangle</p>
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<h3 id="successsolutionsuccess"><Success>Solution</Success></h3>
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<p>Given $z_1+z_2+z_3=0$</p>
<p>$\Rightarrow \quad z_1+z_2+z_3=0$</p>
<p>$\Rightarrow \quad \bar{z}_1+\bar{z}_2+\bar{z}_3=0$</p>
<p>divide by $\left|z_1\right|^2$, we get</p>
<p>$\frac{\bar{z}_1}{z_1 \bar{z}_1}+ \frac{\bar{z}_2}{z_2 \bar{z}_2}+ \frac{\bar{z}_3}{z_3 \bar{z}_3} = 0$</p>
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### <Success>Problem</Success>
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<ul>
<li>Let $z_1, z_2, z_3$ be complex numbers such that</li>
<li>$\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$</li>
<li>$z_1+z_2+z_3 \neq 0$</li>
<li>$z_1^2+z_2^2+z_3^2=0$</li>
</ul>
<p>Then show that for all integer $n \geq 2$</p>
<p>$\left|z_1^n+z_2^n+z_3^n\right| \in\{0,1,2,3\}$</p>
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<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-05-chapter-15-complex-numbers-l-15_15-dqdckzwhm7g/img/155-0828.0.jpg"></p>
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<h3 id="successsolutionsuccess-2"><Success>Solution</Success></h3>
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<p>observation, $z_1^2, z_2^2, z_3^2$ are distinct</p>
<p>$ z_1^2=z_2^2  \Rightarrow \quad z_3^2=-2 z_1^2 \\  \Rightarrow\left|z_3\right|^2=2 \quad \Rightarrow \Leftarrow$</p>
<p>$T\left(z_1^2, z_2^2, z_3^2\right)$ is an equilateral triangle</p>
<p>Note:</p>
<p>$\left|z_1^2\right|=\left|z_2^2\right|=\left|z_3^2\right|=1$</p>

<h3 id="successsolutionsuccess-3"><Success>Solution</Success></h3>
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<p>i.e. $z_2^2=\omega z_1^2 = \omega^4 z_1^2$</p>
<p>$\Rightarrow z_2= \pm \omega^2 z_1$</p>
<p>$z_3^2=\omega^2 z_1^2$</p>
<p>$\Rightarrow z_3= \pm w z_1$</p>
<p>$\left|z_1^n+z_2^n+z_3^n\right|  =\left|z_1^n\left(1 \pm \omega^{2 n} \pm \omega^n\right)\right|$</p>
<p>$\quad \quad =\left|1 \pm \omega^{2 n} \pm \omega^n\right| \leqslant 3 $</p>
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<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-05-chapter-15-complex-numbers-l-15_15-dqdckzwhm7g/img/222-1192.8.jpg"></p>
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<h3 id="successquestionsuccess"><Success>Question</Success></h3>
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<p>$ \omega^{2 n}=\omega^n \Leftrightarrow \omega^n=1$</p>
<p>$ 1 \quad$   $\omega$  $\quad$   $\omega^2$</p>
<p>$1  \quad $  $\omega$   $\quad$   $-\omega^2$</p>
<p>$ 1    \quad $  $-\omega$  $\quad $   $\omega^2$</p>
<p>$ 1  \quad $   $-\omega$    $\quad$  $-\omega^2$</p>
<p>Sum $\in \text{\{0,1,2,3\}}.$</p>
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<h3 id="successproblemsuccess-1"><Success>Problem</Success></h3>
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<p><strong>Compute</strong></p>
<p>$\left(1-w+w^2\right)\left(1-w^2+w^4\right) \cdots\left(1-w^n+w^{2 n}\right)$</p>
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<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-05-chapter-15-complex-numbers-l-15_15-dqdckzwhm7g/img/274-1507.2.jpg"></p>
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### <Success>Solution</Success>
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<p><strong>Recall:</strong>
$1+w+w^2  =0$</p>
<p>$\omega^n= \begin{cases}1 \quad \text { if } n \equiv 0(\operatorname{mod} 3) \\ \omega \quad \text { if } n \equiv 1(\operatorname{mod} 3) \\ \omega^2 \quad \text { if } n \geq 2(\operatorname{mod} 3)\end{cases}$</p>
<p>$1-w+w^2  =-w-w \  =-2 w$</p>
<p>$1-\omega^2+\omega^4=1-\omega^2+\omega=-2 \omega^2$</p>
<p>$1-\omega^3+w^6=1 \\  1-w^n+w^{2n}= \quad \begin{cases} 1  \text { if } n \equiv 0(\bmod 3) \\ -2 \omega  \text { if } n \equiv 1(\bmod 3) \\ -2 \omega^2  \text { if } n \equiv 2(\bmod 3) \end{cases}$</p>
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### <Success>Equation of a line in complex plane</Success>
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<p>Equation of a line in the complex plane is $\bar{\alpha} \bar{z}+\alpha z+\beta=0$</p>
<p>where $0 \neq \alpha \in \mathbb{C}, \beta \in \mathbb{R}$, and $z=x+i y \in \mathbb{C}$.</p>
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### <Success>Equation of a line in complex plane</Success>
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<p>Proof: Ax+By+C = 0</p>
<p>where, A,B,C $\in {\R}$ & $A^2+B^2 \neq 0$</p>
<p>Let $z=x+i y$</p>
<p>$x=\frac{z+\bar{z}}{2} \quad y=\frac{z-\bar{z}}{2 i}  \\ \Rightarrow  A\left(\frac{z+\bar{z}}{2}\right)+B\left(\frac{z-\bar{z}}{2 i}\right)+C=0 \\ \Rightarrow  \quad \bar{z}\left(\frac{A+i B}{2}\right)+z\left(\frac{A-i B}{2}\right)+C=0$</p>

<h3 id="successproblemsuccess-2"><Success>Problem</Success></h3>
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<p><strong>Prove that</strong> $|z-\alpha|^2+|z-\beta|^2=k$ represents a circle iff $|\alpha-\beta|^2<2 k$</p>
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<p><img alt="image" src="http://115.241.75.180/iitpal-images/m11/mathematics-class-11-unit-05-chapter-15-complex-numbers-l-15_15-dqdckzwhm7g/img/551-2974.8.jpg"></p>
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### <Success>Solution</Success>
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<p>$|z-\alpha|^2+|z-\beta|^2=k$</p>
<p>$ (z-\alpha)(\bar{z}-\bar{\alpha})+(z-\beta)(\bar{z}-\bar{\beta})=k \\  2 z \bar{z}-z \bar{\alpha}-\bar{z} \alpha+|\alpha|^2 -z \bar{\beta}-\bar{z} \beta+|\beta|^2=k \\  2 z \bar{z}-z(\bar{\alpha}+\bar{\beta})-\bar{z}(\alpha+\beta)+|\alpha|^2+|\beta|^2-k=0 \\  z \bar{z}-z\left(\frac{\bar{\alpha}+\bar{\beta}}{2}\right)-\bar{z}\left(\frac{\alpha+\beta}{2}\right)+c=0 \leftarrow (*)$</p>
<p>$(*)$ is circle $\Leftrightarrow$  $c < \left|\frac{\alpha + \beta}{2}\right|^2$</p>
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