Question

The region of argand diagram defined by \( |z-1|+|z+1|\leq 4\) is

• A. interior of an ellipse
• B. exterior of a circle
• C. interior and boundary of an ellipse
• D. None of the above

Answer 1

The correct option is C interior and boundary of an ellipse

We have, $|z-1|+|z+1|\le 4$
$\Rightarrow \sqrt{(x-1{)}^2+y^2}+\sqrt{(x+1{)}^2+y^2\le 4}$
$wherez=x+iy$
$\Rightarrow A+B\le 4$
$where,A=\sqrt{(x-1{)}^2+y^2}andB=\frac{....\left(i\right)}{\sqrt{(x+1{)}^2+y^2}}$
But $\left[\right(x-1{)}^2+y^2]-[(x-1{)}^2+y^2\le -x]$
$i.e,A^2-B^2=-4x.......\left(ii\right)$
Dividing Eqs. (ii) by (i), we get
$\sqrt{(x-1{)}^2+y^2}-\sqrt{(x+1{)}^2+y^2}\le -x$
$\Rightarrow 2\sqrt{(x+1{)}^2+y^2}\le 4-x$
$3x^2+4y^2\le 12\left[squaringandsimplifying\right]$
or, $\frac{x^2}{4}+\frac{y^2}{3}\le 1$
which represents the interior and boundary of the ellipse.