Question

Let $zϵC$, the set of complex numbers. Then the equation, $2|z+3i|-|z-i|=0$ represents.

• A. A circle with diameter $\frac{10}{3}$
• B. A circle with radius $\frac{8}{3}$
• C. An ellipse with length of minor axis $\frac{16}{9}$
• D. An ellipse with length of major axis $\frac{16}{3}$

Answer 1

The correct option is B A circle with radius $\frac{8}{3}$

Given:

$\Rightarrow 2|z+3i|-|z-i|=0$ ...............(1)

Substituting $z=x+iy$ in (1),

$\Rightarrow 2|x+i(y+3)|=|x+i(y-1)|$

$\Rightarrow 2\sqrt{x^2+(y+3{)}^2}=\sqrt{x^2+(y-1{)}^2}$

$\Rightarrow 4x^2+4(y+3{)}^2=x^2+(y-1{)}^2$

$\Rightarrow 3x^2=y^2-2y+1-4y^2-24y-36$

$\Rightarrow 3x^2+3y^2+26y+35=0$

$\Rightarrow x^2+y^2+\frac{26}{3}y+\frac{35}{3}=0$

This is the equation of the circle

$\therefore radius=r=\sqrt{0+\frac{169}{9}-\frac{35}{3}}$

$=\sqrt{\frac{64}{9}}=\frac{8}{3}$.