Question

If \( z \) is a complex number and \( k \in \mathbb{R} \), such that \( |z| = 1 \), \[ \frac{2 + k^2 z}{k + \overline{z}} = kz, \] then the maximum distance from \( k + i k^2 \) to the circle \( |z - (1 + 2i)| = 1 \) is:

• A. \( \sqrt{5} + 1 \)
• B. 2
• C. 3
• D. \( \sqrt{5} + \sqrt{1} \)

Hint:

Use geometry and coordinate form of complex numbers to compute distances.

Answer 1

The Correct Option is A

This problem requires finding the value of a real number \( k \) from a given equation involving a complex number \( z \) with modulus 1. Then, we must calculate the maximum distance from the point represented by \( k + ik^2 \) to a given circle in the complex plane.

Concept Used:

1. Properties of Complex Numbers: For any complex number \( z \), its modulus squared is given by \( |z|^2 = z\bar{z} \), where \( \bar{z} \) is the complex conjugate of \( z \). If \( |z| = 1 \), this implies \( z\bar{z} = 1 \).

2. Geometry of Complex Numbers: - The expression \( |z - z_0| \) represents the distance between the points corresponding to the complex numbers \( z \) and \( z_0 \) in the Argand plane. - The equation \( |z - z_0| = r \) describes a circle with center at \( z_0 \) and radius \( r \).

3. Maximum Distance from a Point to a Circle: The maximum distance from a point \( P \) to a circle with center \( C_0 \) and radius \( r \) is the sum of the distance between \( P \) and \( C_0 \) and the radius \( r \). \[ \text{Maximum Distance} = |P - C_0| + r \] This occurs along the line connecting the point and the center, at the point on the circle farthest from \( P \).

Step-by-Step Solution:

We are given two conditions involving a complex number \( z \) and a real number \( k \):

\[ |z| = 1 \quad \text{and} \quad \frac{2+k^2 z}{k+\bar{z}} = kz \]

From the condition \( |z| = 1 \), we know that \( z\bar{z} = |z|^2 = 1 \).

Now, we simplify the second given equation. We start by cross-multiplying:

\[ 2 + k^2 z = kz(k + \bar{z}) \]

Distributing the term \( kz \) on the right-hand side, we get:

\[ 2 + k^2 z = k^2 z + kz\bar{z} \]

We can now substitute the property \( z\bar{z} = 1 \) into the equation:

\[ 2 + k^2 z = k^2 z + k(1) \]

The term \( k^2 z \) appears on both sides of the equation, so it cancels out:

\[ 2 = k \]

Thus, we have found the value of the real number \( k \), which is 2.

The problem asks for the maximum distance from the point represented by the complex number \( k + ik^2 \) to the circle \( |z - (1 + 2i)| = 1 \).

First, we find the complex number representing the point. Substituting \( k = 2 \), we get:

\[ P = k + ik^2 = 2 + i(2^2) = 2 + 4i \]

Next, we identify the properties of the given circle from its equation \( |z - (1 + 2i)| = 1 \). This is a circle with:

• Center \( C_0 = 1 + 2i \)
• Radius \( r = 1 \)

Final Computation & Result:

To find the maximum distance from the point \( P(2+4i) \) to the circle, we first calculate the distance, \( d \), between the point \( P \) and the center of the circle \( C_0(1+2i) \):

\[ d = |P - C_0| = |(2 + 4i) - (1 + 2i)| \] \[ d = |(2 - 1) + (4 - 2)i| = |1 + 2i| \]

The modulus of \( 1 + 2i \) is:

\[ d = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

The maximum distance from the point to the circle is the distance from the point to the center plus the radius:

\[ \text{Maximum Distance} = d + r = \sqrt{5} + 1 \]

Therefore, the maximum distance from \( k + ik^2 \) to the circle is \( \sqrt{5} + 1 \).

Answer 2

Given: \[ \frac{2 + kz}{k + z} = kz \Rightarrow (2 + kz) = kz(k + z) \Rightarrow 2 + kz = k^2z + kz^2 \] Solving and using \( |z| = 1 \Rightarrow zz^* = 1 \) gives \( k = 2 \) Now, find distance between \( (k, k^2) = (2, 4) \) and center \( (1, 2) \): \[ \text{Distance} = \sqrt{(2 - 1)^2 + (4 - 2)^2} = \sqrt{1 + 4} = \sqrt{5} \] Max distance from boundary = \( \sqrt{5} + 1 \)