Complex Numbers
Locus and Geometric Interpretation
Grade 11
Question:
<p>Let <i>z</i><sub>1</sub> and <i>z</i><sub>2</sub> be two distinct complex numbers and <i>z</i> = (1 - <i>t</i>)<i>z</i><sub>1</sub> + <i>tz</i><sub>2</sub>, for some real number <i>t</i> with 0 < <i>t</i> < 1 and <i>i</i> = √−1. If arg(<i>w</i>) denotes the principal argument of a non-zero complex number <i>w</i>, then</p>
<p>(a) |<i>z</i> - <i>z</i><sub>1</sub>| + |<i>z</i> - <i>z</i><sub>2</sub>| = |<i>z</i><sub>1</sub> - <i>z</i><sub>2</sub>|</p>
<p>(b) arg(<i>z</i> - <i>z</i><sub>1</sub>) = arg(<i>z</i> - <i>z</i><sub>2</sub>)</p>
<p>(c) Re\(\frac{z - z_1}{z_2 - z_1}\) = 0</p>
<p>(d) arg(<i>z</i> - <i>z</i><sub>1</sub>) = arg(<i>z</i><sub>2</sub> - <i>z</i><sub>1</sub>)</p>
Step-by-Step Solution
Key Concept: A point z on the line segment joining two points z₁ and z₂ satisfies the property that the sum of distances from z to the endpoints equals the distance between the endpoints.
<p>Since <i>z</i> = (1 - <i>t</i>)<i>z</i><sub>1</sub> + <i>tz</i><sub>2</sub> with 0 < <i>t</i> < 1, the point <i>z</i> lies on the line segment joining <i>z</i><sub>1</sub> and <i>z</i><sub>2</sub>. Therefore, |<i>z</i> - <i>z</i><sub>1</sub>| + |<i>z</i> - <i>z</i><sub>2</sub>| = |<i>z</i><sub>1</sub> - <i>z</i><sub>2</sub>|.</p>
Correct Answer: A