Question

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

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

Answer 1

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

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

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