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Chapter 2: Problem 161

For complex numbers \(z_{1}, z_{2}\) if \(\left|z_{1}\right|=12\) and \(\left|z_{2}-3-4 i\right|=5\) then the minimum value \(\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|\) is (a) 0 (b) 2 (c) 7 (d) 17

Short Answer

Expert verified
The minimum value of \(|z_{1} - z_{2}|\) is 0. (a)

Step by step solution

01

Express the complex numbers and their magnitudes

We are given |z1| = 12 and |z2 - 3 - 4i| = 5. Let z1 = a + bi and z2 = c + di, where a, b, c, and d are real numbers. Thus, we have the following: |a + bi| = 12 |c + di - 3 - 4i|= 5
02

Evaluate conditions based on the given information

Since z1 is 12 units away from the origin in the complex plane, the locus of z1 will be a circle with a radius of 12, centered at the origin. Similarly, since z2 - 3 - 4i = 5, z2 will be 5 units away from the complex number 3+4i. So the locus of z2 will be a circle with a radius of 5, centered at the point (3,4).
03

Find the minimum distance between z1 and z2

To find the minimum value of |z1 - z2|, we need to find the minimum distance between the two circles we found in Step 2. The minimum distance will be on a straight line. The distance between the two centers is: \(d = \sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = 5\) The sum of the radii of the two circles is 12 + 5 = 17. Since the distance between the centers is less than the sum of the radii of the circles, we know that the circles intersect. In this case, the minimum distance between two points on the circles will be when they touch each other at the intersection point, i.e., |z1 - z2| = 0. Thus, the minimum value of |z1 - z2| is 0. The correct answer is (a) 0.

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Most popular questions from this chapter

If \(Z\) is complex number. Then the locus of the point \(Z\) satisfying arg \([(z-i) /(z+i)]=(\pi / 4)\) is a (a) Circle with center \((-1,0)\) and radius \(\sqrt{2}\) (b) Circle with center \((0,0)\) and radius \(\sqrt{2}\) (c) Circle with center \((0,1)\) and radius \(\sqrt{2}\) (d) Circle with center \((1,1)\) and radius \(\sqrt{2}\)

\(w \neq 1\) is cube root of 1 and \(\left(1+w^{2}\right)^{n}=\left(1+w^{4}\right)^{n}\) then the smallest positive value of \(\mathrm{n}\) is (a) 2 (b) 3 (c) 5 (d) 6

The value of \((-i)^{(-1)}=\) (a) \(-(\pi / 2)\) (b) \((\pi / 2)\) (c) \(\mathrm{e}^{-(\pi / 2)}\) (b) \(\mathrm{e}^{(\pi / 2)}\)

If cube root of unity are \(1, w, w^{2}\) then the roots of the equation \((\mathrm{x}-1)^{3}+8=0\) are (a) \(-1,-1,-1\) (b) \(-1,-1+2 \mathrm{w},-1-2 \mathrm{w}^{2}\) (c) \(-1,1+2 \mathrm{w}, 1+2 \mathrm{w}^{2}\) (d) \(-1,1-2 \mathrm{w},+1-2 \mathrm{w}^{2}\)

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
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