If the imaginary number Z satisfies | Z | = 3 and Z / A + A / Z is a real number, find the real number a=

If the imaginary number Z satisfies | Z | = 3 and Z / A + A / Z is a real number, find the real number a=

Let z = 3W, where w = cos α + isin α, then 1 / w = cos α - isin α,
So Z / A + A / z = (3 / a) * (COS α + isin α) + (A / 3) * (COS α - isin α)
=(3/a+a/3)*cosα+i(3/a-a/3)sinα
Since Z / A + A / Z is a real number, then (3 / A-A / 3) sin α = 0
If Z is an imaginary number, then sin α≠ 0, then 3 / A-A / 3 = 0, so a = ± 3

Given that the module of imaginary number (X-2) + Yi (x, y ∈ R) is √ 3, how to find the value range of (y + 1) / (x + 1)?
(X-2) ^ 2 + y ^ 2 = 3 is obtained from the module of "(X-2) + Yi (x, y ∈ R) as √ 3, that is, the circle with (2,0) as the center and √ 3 as the radius. Let (y + 1) / (x + 1) = k, that is, find the value range of K when the line (y + 1) / (x + 1) = k has an intersection with the circle. I know that the straight line (y + 1) / (x + 1) = k must pass through the point (- 1, - 1), but when the line is tangent to the circle, how to calculate this k?

There's something in common
The distance from the center of a circle to a straight line is less than or equal to the radius
y+1=kx+k
kx-y+k-1=0
|2k-0+k-1|/√(k^2+1)

Given that the module of imaginary number (A-2) + bi is √ 3, then the maximum value of B / A is
Help. Thank you

This problem is equivalent to (X-2) ^ 2 + y ^ 2 = 3, then you can draw a graph to compare the maximum value of Y / X. (X-2) ^ 2 + y ^ 2 = 3, which represents a circle in the coordinate system. The center of the circle is m (2,0) x, y, which satisfies (X-2) ^ 2 + y ^ 2 = 3. In fact, the point P (x, y) is a moving point on the circle. What you want to find is the maximum value of Y / x, and write it as (y-0) / (x-0)