Question

If z${}_1$ and z${}_2$ are lying on $|z-3|\le 4$ and $|z-1|+|z+1|=3$ respectively then which of the following integers lie in the range of $|z_1-z_2|$.

• A. $1$
• B. $0$
• C. $9$
• D. $10$

Answer 1

Answer: (A), (B)

The correct options are
A $1$
B $0$

$z_1$ lies on $|z-3|\le 4$

which represents a circle with center $(3,0)$ and radius $2$

$z_2$ lies on $|z-1|+|z+1|=3$

Assume $z_2=x+iy$

Substitute in the above equation, we get

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

Solving this, we get

$\frac{x^2}{\frac{9}{4}}+\frac{y^2}{\frac{5}{4}}=1$

which represents an ellipse with center $(0,0)$ and $a=\frac{3}{2}$ and $b=\frac{\sqrt{5}}{2}$

Now,

For $|z_1-z_2|$, from figure we get

Minimum is $0$ and Maximum is $\frac{13}{2}$

So, option A and B only.