The region of the Argand plane defined by |z | vedclass

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(C) The given inequality is $|z - 1| + |z + 1| \le 4$.This is of the form $|z - z_1| + |z - z_2| \le 2a$,where $z_1 = 1$ and $z_2 = -1$.The distance b

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Interior and boundary of an ellipse

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(C) The given inequality is $|z - 1| + |z + 1| \le 4$.This is of the form $|z - z_1| + |z - z_2| \le 2a$,where $z_1 = 1$ and $z_2 = -1$.The distance between the foci $z_1$ and $z_2$ is $2c = |1 - (-1)| = 2$,so $c = 1$.The sum of the distances from any point $z$ to the foci is $2a = 4$,so $a = 2$.Since $2a > 2c$,this represents the region consisting of the interior and boundary of an ellipse.Using the relation $b^2 = a^2 - c^2$,we get $b^2 = 2^2 - 1^2 = 3$.The equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,which is $\frac{x^2}{4} + \frac{y^2}{3} = 1$.Thus,the region is the interior and boundary of this ellipse.

The region of the Argand plane defined by $|z - 1| + |z + 1| \le 4$ is

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