Question

If \(z_1=8+4i,\; z_2=6+4i\) and \[ \operatorname{Arg}\left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi}{4}, \] then \(z\) satisfies

• A. \( |z-7-4i|=1 \)
• B. \( |z-7-5i|=\sqrt{2} \)
• C. \( |z-4i|=8 \)
• D. \( |z-1-7i|=\sqrt{18} \) \bigskip

Hint:

The locus \[ \operatorname{Arg}\left(\frac{z-z_1}{z-z_2}\right)=\alpha \] represents the circle through \(z_1\) and \(z_2\) from which the chord \(z_1z_2\) subtends a constant angle \(\alpha\). Use \[ AB=2R\sin\alpha \] to find the radius and then determine the centre from the perpendicular bisector of the chord.

Answer 1

The Correct Option is B

Concept: The given argument represents the angle subtended by a fixed chord at a moving point, so the locus is a circle through the two fixed points.

Step 1: The points are \(A(8,4)\) and \(B(6,4)\), giving \(AB=2\). Since the angle subtended is \(\frac{\pi}{4}\), use \(AB=2R\sin\frac{\pi}{4}\) to obtain \(R=\sqrt2\).

Step 2: The midpoint of \(AB\) is \((7,4)\). The centre lies on the perpendicular bisector \(x=7\). Since \(CM=\sqrt{R^2-\left(\frac{AB}{2}\right)^2}=1\), the possible centres are \((7,5)\) and \((7,3)\).

Step 3: From the given positive argument, the required circle lies above the chord. Hence the centre is \((7,5)\). Therefore, the locus is \(\boxed{|z-(7+5i)|=\sqrt2}\).