Discussion
If the complex number z varies so that the real and imaginary parts of \(z - 2 - 3i\) are equal, then the locus of \(z\) is
Answer: B
The locus is a straight line.
Write \(z = x + iy\). Then \(z - 2 - 3i = (x - 2) + i(y - 3)\), and equal real and imaginary parts means:
\[ \begin{aligned} x - 2 &= y - 3 \\ x - y + 1 &= 0 \end{aligned} \]
This is a first-degree equation in \(x\) and \(y\), so the point \(z\) moves on a line of slope 1 through \((2, 3)\).
A circle, parabola, ellipse or hyperbola needs a second-degree equation; that would arise if the condition involved the modulus of \(z - 2 - 3i\) or a product of its parts, not a plain equality of the two parts.
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