Question

If a is a complex constant such that $a{z}^2+z+\overline{a}=0$ has a real root, then

• A. $a+\overline{a}=1$
• B. $a+\overline{a}=0$
• C. $a+\overline{a}=-1$
• D. the absolute value of the real root is 1

Answer 1

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

$a+\overline{a}=1$
the absolute value of the real root is 1
$a+\overline{a}=-1$

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