Explanation

(c)
Given, $|x + iy + 8| + |x + iy - 8| = 16$
$\Rightarrow \ \ \ \ \ \ \left| (x + 8) + iy \right| = 16 - \left| (x - 8) + iy \right|$
$\Rightarrow \ \ \ \ \ \sqrt{(x + 8)^{2} + y^{2}} = 16 - \sqrt{(x - 8)^{2} + y^{2}}$
$\Rightarrow \ \ \ \ x^{2} + 64 + 16x + y^{2} = 256 + x^{2} + 64$
$- 16x + y^{2} - 32\sqrt{(x - 8)^{2} + y^{2}}$
$\Rightarrow \ \ \ \ 32x = 32\lbrack 8 - \sqrt{(x - 8)^{2} + y^{2}}\rbrack$
$\Rightarrow \ \ \ \ {\sqrt{(x - 8)^{2} + y^{2}}}^{\ } = 8 - x$
$\Rightarrow \ \ \ \ \ \ (x - 8)^{2} + y^{2} = (8 - x)^{2}$
$\Rightarrow \ \ \ y^{2} = 0\ \ \ \ \ \Rightarrow \ \ \ y = 0$
Which, represents a straight line.