Answer:
The value of a is -6.
Step-by-step explanation:
A complex number is defined as
[tex]z=x+iy[/tex]
[tex]|z|=|x+iy|=\sqrt{x^2+y^2}[/tex]
where, x is real part and iy is imaginary part.
The given equation is
[tex]|a-i|=\sqrt{37}[/tex]
Here real part is a and imaginary part is -i. So, x=a and y=-1.
[tex]\sqrt{a^2+(-1)^2}=\sqrt{37}[/tex]
Taking square on both the sides.
[tex]a^2+(-1)^2=37[/tex]
[tex]a^2+1=37[/tex]
Subtract 1 from both the sides.
[tex]a^2=36[/tex]
Taking square root on both the sides.
[tex]a=\pm \sqrt{36}[/tex]
[tex]a=\pm 6[/tex]
The value of a is either 6 or -6. But it is given that the picture is in third quadrant, so the value of a can not be positive.
Therefore the value of a is -6.
