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: Understanding the Question:
The question gives the amplitude (argument angle) of a shifted complex variable expression and asks us to find the geometric Cartesian equation representing the path locus of $z = x + iy$.

Step 2: Key Formula or Approach:
The amplitude or argument of a complex number $Z = R + iI$ is defined by:
$$\text{Amp}(Z) = \tan^{-1}\left(\frac{\text{Imaginary Part}}{\text{Real Part}}\right)$$ We substitute $z = x + iy$ into the expression, isolate the real and imaginary parts, and evaluate using the given angle.

Step 3: Detailed Explanation:
Let's substitute $z = x + iy$ into the complex expression:
$$z - 2 - 3i = (x + iy) - 2 - 3i = (x - 2) + i(y - 3)$$ Identify the component terms:
Real part ($R$) = $x - 2$
Imaginary part ($I$) = $y - 3$
We are given that $\text{Amp}(z - 2 - 3i) = \frac{3\pi}{4}$, so:
$$\tan^{-1}\left(\frac{y - 3}{x - 2}\right) = \frac{3\pi}{4} \implies \frac{y - 3}{x - 2} = \tan\left(\frac{3\pi}{4}\right)$$ We know that $\tan\left(\frac{3\pi}{4}\right) = -1$. Substituting this value gives:
$$\frac{y - 3}{x - 2} = -1$$ Cross-multiply to clear the fraction:
$$y - 3 = -(x - 2)$$ $$y - 3 = -x + 2$$ Rearranging all variables to the left side gives:
$$x + y = 2 + 3 \implies x + y = 5$$ This matches option (B).

Step 4: Final Answer:
The locus of $z$ is the straight line $x + y = 5$, which corresponds to option (B).