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Chapter 2: Problem 146

If the imaginary part of \([(2 z-3) /(i z+1)]\) is \(-2\) then the locus of the point representing \(z\) in the complex plane is (a) a circle (b) a straight line (c) a parabola (d) an ellipse

Short Answer

Expert verified
The locus of the point representing \(z\) in the complex plane is an ellipse. We found this by simplifying the given expression, setting the imaginary part equal to -2, and then recognizing the geometric representation of the resulting equation. So, the correct answer is (d) an ellipse.

Step by step solution

01

Write the given complex expression and define the real(z) and imaginary(z) parts

The given expression is \(\frac{2z-3}{iz+1}\). Let z = a + bi, where a is the real part (Re(z)) and b is the imaginary part (Im(z)) of the complex number z.
02

Substitute z

Replace z with a + bi in the given expression: \[\frac{2(a+bi)-3}{i(a+bi)+1}\]
03

Simplify the expression

Simplify the expression by separating the real and imaginary parts: \[\frac{(2a-3)+2bi}{(1-b)+ai}\]
04

Multiply by conjugate

Multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary part from the denominator: \[\frac{(2a-3)+2bi}{(1-b)+ai} * \frac{(1-b)-ai}{(1-b)-ai} = \frac{((2a-3)(1-b) - 2ab) + (2b+(2a-3)a)i}{(1-b)^2 + a^2}\]
05

Set the imaginary part equal to -2

Now, the imaginary part of the expression is given as -2. Equate the imaginary part on the left side of the expression to -2: \[\frac{2b+(2a-3)a}{(1-b)^2 + a^2} = -2\]
06

Remove the fraction

Remove the fraction by multiplying both sides with \(((1-b)^2 + a^2)\): \[2b+(2a-3)a = -2((1-b)^2 + a^2)\]
07

Simplify and solve

Simplify the equation and collect like terms: \[a^2 + 2b^2 - 4a + 6b + 2 = 0\]
08

Recognize the equation's geometric representation

From the resulting equation, we notice that there is a second-degree term for both the real and imaginary parts, so the locus of the point representing z (Re(z) and Im(z)) is an ellipse. So, the correct answer is (d) an ellipse.

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Most popular questions from this chapter

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are integers, not all equal, and \(\mathrm{w}\) is a cube root of unity \((\mathrm{w} \neq 1)\) Then the minimum value of \(\left|\mathrm{a}+\mathrm{bw}+\mathrm{cw}^{2}\right|\) is (a) 0 (b) 1 (c) \((\sqrt{3} / 2)\) (d) \((1 / 2)\)

The equation \(|z-i|+|z+i|=k\) represent an ellipse if \(K=\) (a) 1 (b) 2 (c) 4 (d) \(-1\)

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For any integer \(\mathrm{n}, \arg \left[(\sqrt{3}+\mathrm{i})^{4 n+1} /(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}\right]=\) (a) \((\pi / 3)\) (b) \((\pi / 6)\) (c) \((2 \pi / 3)\) (d) \((5 \pi / 6)\)

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