Question

The area of the triangle on the complex plane formed by the points z, z+iz and iz is 128. Then the value of |z| is

• A. 12
• B. 16
• C. 18
• D. 17
• E. 19

Answer 1

The Correct Option is B

We are given that the area of the triangle formed by the points \(z, z + iz, iz\) on the complex plane is 128, and we need to find \(|z|\).

The vertices of the triangle are represented by the points: \(z \), \( z + iz \), and \( iz\). 

To find the area of the triangle, we use the formula for the area of a triangle in the complex plane given by the points \(A = z_1 \), \( B = z_2 \), \( C = z_3\):

\(\text{Area} = \frac{1}{2} \left| \text{Im}(z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}) \right|\)

Here, the points are \(A = z \), \( B = z + iz \), and \( C = iz \), so we need to calculate the complex number\)

\(A = z, B = z + iz, C = iz\)

The area can also be calculated using the formula for the area of a triangle with vertices at complex numbers \(A = z \), \( B = z + iz \), \( C = iz\):

\(\text{Area} = \frac{1}{2} | \text{Im}(z \cdot (z + iz) - z \cdot iz) |\)

By solving, we find that the area of the triangle is related to the value of \(|z|\), and it is given that the area is 128:

\(\frac{1}{2} | \text{Imaginary part} | = 128\)

Solving for \(|z|\), we find that \(|z| = 16\).

The answer is 16.