Answer:
a) The minimum value of modulus of n is
[tex] \sqrt{10} [/tex]
b) An example of complex number n is
[tex]3 + i[/tex]
Step-by-step explanation:
The given complex number is
[tex]m = 2 + 6i[/tex]
and the given inequality is
[tex] |m + n| = 3 \sqrt{10} [/tex]
By the triangle inequality property:
[tex] |m + n| \geqslant |m| + |n| [/tex]
This implies that:
[tex] |m| + |n| \geqslant 3 \sqrt{10} [/tex]
[tex] |2 +6i| + |n| \geqslant 3 \sqrt{10} [/tex]
[tex] \sqrt{ {2}^{2} + {6}^{2} } + |n| \geqslant 3 \sqrt{10}[/tex]
[tex] \sqrt{ 4 +36 } + |n| \geqslant 3 \sqrt{10}[/tex]
[tex] \sqrt{ 40 } + |n| \geqslant 3 \sqrt{10}[/tex]
[tex]2\sqrt{10 } + |n| \geqslant 3 \sqrt{10}[/tex]
[tex]|n| \geqslant 3 \sqrt{10} - 2 \sqrt{10} [/tex]
[tex] |n| \geqslant \sqrt{10}[/tex]
The minimum value of the modulus of n is √10
b) Let n=a+bi
[tex] |n| \geqslant \sqrt{10} \\ \implies |a + bi| = \sqrt{10} [/tex]
[tex] \sqrt{ {a}^{2} + {b}^{2} } = \sqrt{10} [/tex]
[tex] {a}^{2} + {b}^{2} = 10[/tex]
[tex] {a}^{2} = 10 - {b}^{2} [/tex]
[tex] {a} = \pm \sqrt{10 - {b}^{2} } [/tex]
when b=1,
[tex]{a} = \pm \sqrt{10 - {1}^{2} } \\ {a} = \pm \sqrt{9} \\ {a} = \pm 3 \\ a = - 3 \: or \: a = 3[/tex]
Therefore one example of complex number n is:
[tex]n = 3 + i[/tex]
