Question

Locus of complex number satisfying $\arg \left[\frac{(z-5+4i)}{(z+3-2i)}\right]=-\frac{\pi }{4}$ is the arc of a circle

• A. whose radius is $5\sqrt{2}$ units
• B. whose radius is $5$ units
• C. whose length (of arc) is $\frac{15\pi }{\sqrt{2}}$ units
• D. whose centre is $-2-5i$

Answer 1

Answer: (A), (C), (D)

The correct options are
A whose radius is $5\sqrt{2}$ units
C whose length (of arc) is $\frac{15\pi }{\sqrt{2}}$ units
D whose centre is $-2-5i$


$\frac{z_0-(-3+2i)}{z_0-(5-4i)}=\frac{BD}{AD}{e}^{i\pi /2}=i$
$\Rightarrow z_0+3-2i=i{z}_0-5i-4$
$\Rightarrow z_0=-2-5i$
$\Rightarrow$ Radius $AD=|5-4i-(-2-5i)|$
$=|7+i|$
$=\sqrt{50}=5\sqrt{2}$ units
Length of arc $=\frac{3}{4}$ (perimeter of circle)
$=\frac{3}{4}(2\pi \times 5\sqrt{2})$
$=\frac{15\pi }{\sqrt{2}}$ units