Question

Let $z \in C$, the set of complex numbers. Then the equation, $2 | z + 3i| - | z - i| = 0 $ represents :

• A. a circle with radius $\frac{8}{3}$
• B. a circle with diameter $\frac{10}{3}$
• C. an ellipse with length of major axis $\frac{16}{3}$
• D. an ellipse with length of minor axis $\frac{16}{9}$

Answer 1

The Correct Option is A

 To solve the problem, let's first analyze the given equation:

\(2 | z + 3i| - | z - i| = 0\)

This equation can be rewritten as:

\(2 | z + 3i| = | z - i|\)

This implies:

\(| z + 3i| = \frac{1}{2} | z - i|\)

The expression \(| z + a|\) denotes the distance of \(z\) from a point \(-a\) in the complex plane. Thus, \(| z + 3i|\) is the distance from the point \((-3i)\) and \(| z - i|\) is the distance from the point \((i)\).

The given equation can be interpreted as a geometric locus where the distance from \((-3i)\) is half of the distance from \((i)\). Such a locus is a circle.

Now, let's calculate the center and radius of this circle.

The distance between these two points is \(| -3i - i | = |-4i| = 4\).

Given the general form of this relation \(| z - (foci_1)| = e \cdot | z - (foci_2)|\) where \(e\) is the eccentricity.

Comparing with the standard form \(|z - f_1| = k \cdot |z - f_2|\), here \(k = \frac{1}{2}\). The locus is a circle if \(0 < k < 1\).

The center of the circle can be calculated as the weighted average of the two foci:

\(\text{Center} = \frac{f_1 + f_2}{1+e} = \frac{(-3i) + i}{1 + \frac{1}{2}} = \frac{-2i}{\frac{3}{2}} = -\frac{4}{3}i\)

The radius of the circle is given by:

\(\frac{4}{1 + k} = \frac{4}{1 + \frac{1}{2}} = \frac{8}{3}\)

Thus, the equation represents a circle with center \(- \frac{4}{3} i\)and radius \(\frac{8}{3}\).

Therefore, the correct answer is:

a circle with radius \(\frac{8}{3}\)