Complex Numbers
Complex Plane / Geometry
Grade Class 11
Question:
<p>Let \( z_1 = 10+6i,\ z_2 = 4+6i \) and \( \arg\left(\dfrac{z-z_1}{z-z_2}\right) = \dfrac{\pi}{4} \). Then:</p>
\(|z-7-9i| = 3\sqrt{2}\)
\(|z-7-9i| = 5\)
\(\text{Re}(z) > 7\)
\(\text{Im}(z) > 9\)
Step-by-Step Solution
Key Concept: The locus of z with arg((z-z_1)/(z-z_2)) = \pi/4 is an arc of a circle. The centre and radius can be determined from the chord z_1z_2.
<p>The midpoint of \(z_1 z_2\) is \(7+6i\). The centre of the circle is at \(7+9i\) (perpendicular bisector shifted by |z_1-z_2|/2 \cdot cot(\pi/4) = 3). Radius = \(3\sqrt{2}\). Both A and B hold for the arc.</p>
Correct Answer: AB