Question

Let $z_1$ satisfy the condition $|z-3|=2$ and $z_2$ satisfy the equation $|z-1|+|z+1|=3$. If $|z_1-z_2{|}_{min}=m$ and $|z_1-z_2{|}_{max}=M$, then which of the following is/are correct?

• A. $M+m=\frac{9}{2}$
• B. $M^2+m^2+Mm=\frac{81}{4}$
• C. $M^2+m^2+Mm=\frac{169}{4}$
• D. $M+m=\frac{13}{2}$

Answer 1

Answer: (C), (D)

$M+m=\frac{13}{2}$

$|z-1|+|z+1|=3$

Ellipse: Centre $(0,0),2a=3,2ae=2$

$|z-3|=2\to$ Circle with centre $(3,0),r=2$



$m=|z_1-z_2{|}_{min}=0$ as curves intersect.

$|z_1-z_2{|}_{max}=5+\frac{3}{2}=\frac{13}{2}=M$