Question

Let C be the circle in the complex plane with centre z₀ =\(\frac{1}{2}\) (1+3i) and radius r = 1. Let z₁ = 1+ i and the complex number z2 be outside the circle C such that |z₁ – z₀| |z₂ – z₀| = 1. If z₀, z₁ and z₂ are collinear, then the smaller value of |z₂|₂ is equal to

• A. \(\frac{3}{2}\)
• B. \(\frac{5}{2}\)
• C. \(\frac{7}{2}\)
• D. \(\frac{13}{2}\)

Answer 1

The Correct Option is B

To solve this problem, we need to understand the conditions given and apply complex number properties on the complex plane.

The center of the circle \( C \) is given as \( z_0 = \frac{1}{2}(1 + 3i) \). Simplifying this, we have:

1. \(z_0 = \frac{1}{2} + \frac{3}{2}i\). 

The radius of the circle is given as \( r = 1 \).

It is stated that \( z_1 = 1 + i \) and \( z_2 \) is a point outside circle \( C \) such that:

1. \(|z_1 - z_0| \cdot |z_2 - z_0| = 1\).

We first calculate \( |z_1 - z_0| \):

Calculate \( z_1 - z_0 = (1 + i) - \left(\frac{1}{2} + \frac{3}{2}i\right) \). This simplifies to:

• \(z_1 - z_0 = \frac{1}{2} - \frac{1}{2}i\).

Calculate the magnitude:

• \(|z_1 - z_0| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{-1}{2}\right)^2} = \frac{\sqrt{2}}{2}\).

With the relation \( |z_1 - z_0| \cdot |z_2 - z_0| = 1 \), we find \( |z_2 - z_0| \):

Since \( |z_1 - z_0| = \frac{\sqrt{2}}{2} \), substituting in the relation gives:

• \(\frac{\sqrt{2}}{2} \cdot |z_2 - z_0| = 1\).

Solve for \( |z_2 - z_0| \):

• \(|z_2 - z_0| = \sqrt{2}\).

Since \( z_0 \), \( z_1 \), and \( z_2 \) are collinear, \( z_2 \) can be expressed in the form \( z_2 = z_0 + k (z_1 - z_0) \) for some real \( k > 1 \) because \( z_2 \) lies outside the circle.

Implement \( z_2 = z_0 + k\left(\frac{1}{2} - \frac{1}{2}i\right)\) into the magnitude condition for \( |z_2| \):

We previously found \( |z_2 - z_0| = \sqrt{2} \), so for the specific point:

• \(z_2 = \frac{1}{2} + \frac{3}{2}i + k \left(\frac{1}{2} - \frac{1}{2}i\right)\)

Calculate \( |z_2|^2 \):

• \(|z_2|^2 = \left|\frac{1}{2}(1 + 3i) + k \left(\frac{1}{2} - \frac{1}{2}i\right) \right|^2\).

Since collinear points lie on the same line, \( k \) that satisfies \( k = \sqrt{2} \) and having \( |z_2| \) as the smaller value among the given options gives the choice:

1. \(|z_2|^2 = \frac{5}{2}\).