Question

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

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

Answer 1

The Correct Option is A

The correct answer is A: a circle with radius \(\frac{8}{3}\)
Given that;
The equation is :\(z|z+3i|-|z-i|=0-(i)\)
Now substitute :\(z=x+iy\) in equation (i)
\(\therefore\)\(2\left|x+i\left(y+3\right)\right| = \left|x+i \left(y-1\right)\right|\)
\(\Rightarrow 2\sqrt{x^{2}+\left(y+3\right)^{2}} = \sqrt{x^{2}+\left(y-1\right)^{2}}\)
\(\Rightarrow 4x^{2} + 4\left( y + 3\right)^{2} = x^{2} + \left( y - 1\right)^{2}\)
\(\Rightarrow 3x^{2} = y^{2} - 2y + 1 - 4y^{2} - 24y - 36\)
\(\Rightarrow 3x^{2} + 3y^{2} + 26y + 35 = 0\)
\(\Rightarrow x^{2} + y^{2}+\frac{26}{3}y + \frac{35}{3} = 0\)
\(\Rightarrow r = \sqrt{0+\frac{169}{9}-\frac{35}{3}}\)
\(\Rightarrow \sqrt{\frac{64}{9}} = \frac{8}{3}\)