Question

The equation$|z-i|+|z+i|=k,k>0$, can represent an ellipse if k is

• A. 1
• B. 2
• C. 4
• D. none of thees

Answer 1

The correct option is C 4

Let $z=x+iy$
Therefore
$|z-i|+|z+i|=k$
$\sqrt{x^2+(y-1{)}^2}+\sqrt{x^2+(y+1{)}^2}=k$
$\sqrt{x^2+(y-1{)}^2}=k-\sqrt{x^2+(y+1{)}^2}$
$x^2+(y-1{)}^2=k^2+x^2+(y+1{)}^2-2k\sqrt{x^2+(y+1{)}^2}$
$2k\sqrt{x^2+(y+1{)}^2}=k^2+(y+1{)}^2-(y-1{)}^2$
$2k\sqrt{x^2+(y+1{)}^2}=k^2+4y$
$4k^2(x^2+(y+1{)}^2)=k^4+16y^2+8k^{2}y$
$4k^2{x}^2+4k^2{y}^2+8k^{2}y+4k^2=k^4+16y^2+8k^{2}y$
$4k^2{x}^2+y^2(4k^2-16)+4k^2-k^4=0$
Therefore it represents an ellipse if
$4k^2-16>0$
Or
$4k^2>16$
$k^2>4$
$|k|>2$

so $k=4$ is correct option