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Chapter 2: Problem 156

\(\mathrm{A}\left(\mathrm{z}_{1}\right), \mathrm{B}\left(\mathrm{z}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{z}_{3}\right)\) are vertices of \(\triangle \mathrm{ABC}\) where \(\mathrm{m}\) \(\angle \mathrm{C}=(\pi / 2)\) and \(\mathrm{AC}=\mathrm{BC}, \mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}\) are complex number if \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}=\mathrm{k}\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) then \(\mathrm{k}=\) (a) 1 (b) 2 (c) 4 (d) non of these

Short Answer

Expert verified
The value of \(k\) is 2. Hence, the correct option is (b) 2.

Step by step solution

01

Understanding the Problem

We are given a triangle ABC where AC = BC and angle C is 90 degrees. This means ABC is an isosceles right triangle. We also know the complex values of points A, B, and C. The equation \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}=\mathrm{k}\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) is given and we are expected to value of k.
02

Using Complex Analysis Facts

We know from complex analysis that if AB = AC in triangle ABC, then \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2} = 2\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\), where z1, z2, and z3 are the complex representations of points A, B, and C respectively.
03

Substituting the Values

Substituting \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}\) for \(2\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) in the equation, we get the equality \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2} = k\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\).
04

Solving for k

Therefore, by comparing we get the value of k = 2. Hence the correct option is (b) 2. By following the proof, we've learned that the value of \( k \) is 2. If we compare this to the options, we can observe that option B is the correct choice.

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Most popular questions from this chapter

If \(|z-4 / z|=2\) then the maximum value of \(|z|\) is (a) \(\sqrt{5}+1\) (b) 2 (c) \(2+\sqrt{2}\) (d) \(\overline{\sqrt{3}+1}\)

If \(\mathrm{x}=\mathrm{a}+\mathrm{b}, \mathrm{y}=\mathrm{a} \alpha+\mathrm{b} \beta\) and \(\mathrm{z}=\mathrm{a} \beta+\mathrm{b} \alpha\), Where \(\alpha, \beta \neq 1\) are cube roots of unity, then \(\mathrm{xyz}=\) (a) \(2\left(\mathrm{a}^{3}+\mathrm{b}^{3}\right)\) (b) \(2\left(a^{3}-b^{3}\right)\) (c) \(\mathrm{a}^{3}+\mathrm{b}^{3}\) (d) \(a^{3}-b^{3}\)

If \(z=x-\) iy and \(z^{(1 / 3)}=p+i\) q then \(\left[\\{(x / p)+(y / q)\\} /\left(p^{2}+q^{2}\right)\right]\) (a) 2 (b) \(-1\) (c) 1 (d) \(-2\)

The small positive integer ' \(n\) ' for which \((1+i)^{2 n}=(1-i)^{2 n}\) is \(\begin{array}{lll}\text { (a) } 4 & \text { (b) } 8 & \text { (c) } 2\end{array}\) (d) 12

The number of complex numbers \(z\) such that \(|z-1|\) \(=|z+1|=|z-i|\) equal (a) 0 (b) 1 (c) 2 (d) \(\infty\)
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