Question

Locus of the point $z$ satisfying the equation $|iz-1|+|z-i|=2$ is-

• A. straight line segment
• B. a circle
• C. an ellipse
• D. a pair of straight line

Answer 1

The correct option is A straight line segment

$z=x+iy$

$|iz-1|+|z-i|=2$

$|i(x+iy)-1|+|x+iy-i|=2$

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

$\sqrt{(y-1{)}^2+(x{)}^2}+\sqrt{x^2+(y-1{)}^2}=2$ ….$\left(1\right)$

Also, $x^2+(y+1{)}^2-x^2+(y-1{)}^2=4y$ ….$\left(2\right)$

Divide equation $\left(2\right)$ by $\left(1\right)$ we get

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

Adding $\left(1\right)$ and $\left(3\right)$ gives

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

$x^2+(y+1{)}^2=(y+1{)}^2$

$\Rightarrow x=0$ which is y-axis

A straight line (Answer).