First slide

Geometry of complex numbers

Question

The region represented by _{$| z + a | + | z - a | < 4 a$} is

Moderate

A

_{$x^{2} + y^{2} < 4 a^{2}$}
B

_{$x^{2} + y^{2} > 4 a^{2}$}
C

_{$3 x^{2} - 4 y^{2} < 4 a^{2}$}
D

_{$3 x^{2} + 4 y^{2} < 12 a^{2}$}

Solution

We have $s (- a, 0) s^{1} (a, 0)$
$\Rightarrow 4 a e = 2 a$
$\Rightarrow e = \frac{1}{2}$
$∴ t h e l o c u s o f z = x + i y i s \frac{x^{2}}{4 a^{2}} + \frac{y^{2}}{3 a^{2}} < 1$ $(∵ b^{2} = 4 a^{2} (1 - \frac{1}{4}))$
$\Rightarrow 3 x^{2} + 4 y^{2} < 12 a^{2}$

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