SATHEE: Complex Numbers 4 Question 12

Complex Numbers 4 Question 12

12. If one of the vertices of the square circumscribing the circle $| z - 1 | = \sqrt{2}$ is $2 + \sqrt{3} i$ . Find the other vertices of square.

(2005, 4M)

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Answer:

Correct Answer: 12. $z_{2} = - \sqrt{3} i, z_{3} = (1 - \sqrt{3}) + i$ and $z_{4} = (1 + \sqrt{3}) - i$

Solution:

Here, centre of circle is $(1, 0)$ is also the mid-point of diagonals of square

$\begin{array}{lc} \Rightarrow & \frac{z_{1} + z_{2}}{2} = z_{0} \\ \Rightarrow & z_{2} = - \sqrt{3} i [where, z_{0} = 1 + 0 i] \\ and & \frac{z_{3} - 1}{z_{1} - 1} = e^{\pm i π / 2} \\ \Rightarrow & z_{3} = 1 + (1 + \sqrt{3} i) \cdot \cos \frac{π}{2} \pm i \sin \frac{π}{2} [∵ z_{1} = 2 + \sqrt{3} i] \\ = 1 \pm i (1 + \sqrt{3} i) = (1 \mp \sqrt{3}) \pm i = (1 - \sqrt{3}) + i \end{array}$

and $z_{4} = (1 + \sqrt{3}) - i$

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Complex Numbers 4 Question 12

12. If one of the vertices of the square circumscribing the circle |z−1|=2 is 2+3i. Find the other vertices of square.

Answer:

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12. If one of the vertices of the square circumscribing the circle $| z - 1 | = \sqrt{2}$ is $2 + \sqrt{3} i$ . Find the other vertices of square.