Question

If the amplitude of $(z - 2 - 3i)$ is $\frac{3\pi}{4}$, then the locus of $z$ is (where $z = x + iy$)

• A. $x + y = 1$
• B. $x + y = 5$
• C. $x - y = -5$
• D. $x - y = 1$

Hint:

An amplitude equation of the form $\text{Amp}(z - z_0) = \theta$ represents a straight ray extending from the point $z_0$. Since $z_0 = (2, 3)$ and $\theta = 135^\circ$ (slope $-1$), the line equation is simply $y - 3 = -1(x - 2) \implies x + y = 5$ in a single step!

Answer 1

The Correct Option is B

Step 1: Substitute $z = x + iy$.
$z - 2 - 3i = (x-2) + i(y-3)$.
Step 2: Identify the real and imaginary parts.
Real part $= x - 2$ and imaginary part $= y - 3$.
Step 3: Use the meaning of amplitude.
The amplitude is the angle whose tangent is (imaginary part)/(real part): $\tan\theta = \dfrac{y-3}{x-2}$, with $\theta = \dfrac{3\pi}{4}$.
Step 4: Evaluate the tangent of the given angle.
$\tan\dfrac{3\pi}{4} = -1$, so $\dfrac{y-3}{x-2} = -1$.
Step 5: Clear the fraction.
$y - 3 = -(x - 2) = -x + 2$.
Step 6: Rearrange into the line equation.
$x + y = 2 + 3 = 5$, giving the straight line $x + y = 5$.
\[ \boxed{x + y = 5} \]