Question

A complex number z=3+4i is rotated about another fixed complex number z₁=1+2i in anticlockwise direction by 45° angle. Find the complex number represented by new position of z in argand plane

A. 1 + (2 - √2)i
B. 1 + (2 + √2)i
C. 1 - (2 + √2)i
D. 1 + (2 + √2)i

Answer 1

The complex number represented by the new position of z after rotating about z₁ is 1 + (2 + √2)i.

Step-by-step explanation:

To find the complex number represented by the new position of z after rotating about z₁, we need to perform the rotation operation. The rotation formula states that rotating a complex number z by an angle θ about a fixed complex number z₁ is given by:

z' = z₁ + e^(iθ) * (z - z₁)

In this case, z = 3 + 4i, z₁ = 1 + 2i, and θ = 45°. Plugging in these values, we get:

z' = (1 + 2i) + e^(iπ/4) * (3 + 4i - 1 - 2i)

Simplifying this expression, we get:

z' = (1 + 2i) + e^(iπ/4) * (2 + 2i)

Using Euler's formula, e^(iπ/4) = cos(π/4) + i * sin(π/4) = (√2)/2 + (√2)/2 * i

Plugging in this value, we get:

z' = (1 + 2i) + (√2)/2 + (√2)/2 * i)

Simplifying further, we get:

z' = 1 + (√2)/2 + 2i + (√2)/2 * i

Combining like terms, we get:

z' = 1 + (2 + √2)i

Therefore, the complex number represented by the new position of z in the Argand plane is 1 + (2 + √2)i.