Q.Match the statements of Column A and Column B. Column A:

The key idea is Complex Number Arithmetic and Geometry, involving modulus, argument, polar form, loci, and regions in the complex plane.

(a) The polar form of $i+\sqrt{3}$ is:

1. Let $z = \sqrt{3}+i$. The modulus is $|z| = \sqrt{(\sqrt{3})^2+1^2} = \sqrt{3+1} = 2$.
2. Since $z$ is in the first quadrant, the argument is $\arg(z) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$.
3. The polar form is $2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$. Matches (v).

(b) The amplitude of $-1+\sqrt{-3}$ is:

1. Let $z = -1+\sqrt{3}i$. The real part is $-1$ and the imaginary part is $\sqrt{3}$, placing $z$ in the second quadrant.
2. The reference angle is $\alpha = \tan^{-1}\left|\frac{\sqrt{3}}{-1}\right| = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}$.
3. The amplitude (argument) is $\theta = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$. Matches (iii).

(c) If $|z+2|=|z-2|$, then locus of $z$ is:

1. The equation $|z-z_1| = |z-z_2|$ represents the perpendicular bisector of the line segment joining $z_1$ and $z_2$.
2. Here, $z_1 = -2$ (point $(-2,0)$) and $z_2 = 2$ (point $(2,0)$).
3. The locus is the perpendicular bisector of the segment joining $(-2,0)$ and $(2,0)$. Matches (i).

(d) If $|z+2i|=|z-2i|$, then locus of $z$ is:

1. The equation $|z-z_1| = |z-z_2|$ represents the perpendicular bisector of the line segment joining $z_1$ and $z_2$.
2. Here, $z_1 = -2i$ (point $(0,-2)$) and $z_2 = 2i$ (point $(0,2)$).
3. The locus is the perpendicular bisector of the segment joining $(0,-2)$ and $(0,2)$. Matches (iv).

(e) Region represented by $|z+4i|\geq3$ is:

1. This inequality is of the form $|z-z_0| \geq r$.
2. Here, $z_0 = -4i$, which corresponds to the point $(0,-4)$, and the radius $r=3$.
3. The region is on or outside the circle having centre at $(0,-4)$ and radius 3. Matches (ii).

(f) Region represented by $|z+4|\leq3$ is:

1. This inequality is of the form $|z-z_0| \leq r$.
2. Here, $z_0 = -4$, which corresponds to the point $(-4,0)$, and the radius $r=3$. …

Concept and Intuition

Complex numbers extend the real number system by including the imaginary unit $i$, where $i^2 = -1$. They can be represented in several forms:

• Rectangular form: $z = x+iy$, where $x$ is the real part and $y$ is the imaginary part.
• Polar form: $z = r(\cos\theta + i\sin\theta)$, where $r = |z|$ is the modulus (distance from origin) and $\theta = \arg(z)$ is the argument or amplitude (angle with the positive real axis).

Understanding these forms is crucial for operations like finding reciprocals or conjugates, and for interpreting geometric properties.

Geometric Interpretation of Modulus:

• $|z|$ represents the distance of the complex number $z$ from the origin $(0,0)$ in the Argand plane.
• $|z-z_0|$ represents the distance of $z$ from a fixed complex number $z_0$.
• Locus:
  • $|z-z_1| = |z-z_2|$ means $z$ is equidistant from $z_1$ and $z_2$. This describes the perpendicular bisector of the line segment joining $z_1$ and $z_2$.
  • $|z-z_0| = r$ describes a circle centered at $z_0$ with radius $r$.
  • $|z-z_0| \leq r$ describes the region on or inside the circle centered at $z_0$ with radius $r$.
  • $|z-z_0| \geq r$ describes the region on or outside the circle centered at $z_0$ with radius $r$.

Complex Number Arithmetic:

• Conjugate: For $z = x+iy$, its conjugate is $\bar{z} = x-iy$. Geometrically, it's a reflection across the real axis.
• Reciprocal: For $z = x+iy$, its reciprocal is $\frac{1}{z} = \frac{1}{x+iy}$. To simplify, multiply the numerator and denominator by the conjugate of the denominator: $\frac{1}{x+iy} \times \frac{x-iy}{x-iy} = \frac{x-iy}{x^2+y^2}$.

We will apply these concepts to each statement in Column A.

Step-by-step Solution

(a) The polar form of $i+\sqrt{3}$ is:

1. Identify the complex number: The given complex number is $z = \sqrt{3} + i$.
2. Find the modulus ($r$): The modulus is the distance from the origin. $r = |z| = \sqrt{(\text{Re}(z))^2 + (\text{Im}(z))^2} = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3+1} = \sqrt{4} = 2$.
3. Find the argument ($\theta$): The argument is the angle with the positive real axis. Since the real part $\sqrt{3}$ is positive and the imaginary part $1$ is positive, $z$ lies in the first quadrant. $\tan\theta = \frac{\text{Im}(z)}{\text{Re}(z)} = \frac{1}{\sqrt{3}}$. Therefore, $\theta = \frac{\pi}{6}$ (or $30^\circ$).
4. Write in polar form: The polar form is $r(\cos\theta + i\sin\theta)$. $z = 2\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)$. This matches with (v).

(b) The amplitude of $-1+\sqrt{-3}$ is:

1. Simplify the complex number: $\sqrt{-3} = \sqrt{3 \times (-1)} = \sqrt{3}i$. So, the complex number is $z = -1 + i\sqrt{3}$.
2. Identify real and imaginary parts: $\text{Re}(z) = -1$, $\text{Im}(z) = \sqrt{3}$.
3. Determine the quadrant: Since the real part is negative and the imaginary part is positive, $z$ lies in the second quadrant.
4. Find the reference angle ($\alpha$): The reference angle is $\alpha = \tan^{-1}\left(\frac{|\text{Im}(z)|}{|\text{Re}(z)|}\right) = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$.
5. Calculate the amplitude ($\theta$): For a complex number in the second quadrant, the amplitude is $\theta = \pi - \alpha$. $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$. This matches with (iii).

(c) If $|z+2|=|z-2|$, then locus of $z$ is:

1. Interpret the equation geometrically: The equation $|z-z_1| = |z-z_2|$ means that the distance from $z$ to $z_1$ is equal to the distance from $z$ to $z_2$. Here, $z_1 = -2$ (which corresponds to the point $(-2, 0)$ in the Argand plane) and $z_2 = 2$ (which corresponds to the point $(2, 0)$).
2. Identify the locus: The locus of points equidistant from two fixed points is the perpendicular bisector of the line segment joining those two points. The segment joins $(-2, 0)$ and $(2, 0)$. The midpoint is $\left(\frac{-2+2}{2}, \frac{0+0}{2}\right) = (0,0)$. The segment lies on the real axis. Its perpendicular bisector is the imaginary axis, which has the equation $x=0$. This matches with (i).

(d) If $|z+2i|=|z-2i|$, then locus of $z$ is:

1. Interpret the equation geometrically: Similar to part (c), this equation means $z$ is equidistant from $z_1 = -2i$ and $z_2 = 2i$. In the Argand plane, $z_1 = -2i$ corresponds to $(0, -2)$ and $z_2 = 2i$ corresponds to $(0, 2)$.
2. Identify the locus: The locus is the perpendicular bisector of the line segment joining $(0, -2)$ and $(0, 2)$. The midpoint is $\left(\frac{0+0}{2}, \frac{-2+2}{2}\right) = (0,0)$. The segment lies on the imaginary axis. Its perpendicular bisector is the real axis, which has the equation $y=0$. This matches with (iv).

(e) Region represented by $|z+4i|\geq3$ is:

1. Rewrite the inequality: The inequality is $|z - (-4i)| \geq 3$. …