If the complex Z satisfies z = (1 + Ti) / (1-ti) (t ∈ R), the trajectory equation of point Z corresponding to Z is obtained

If the complex Z satisfies z = (1 + Ti) / (1-ti) (t ∈ R), the trajectory equation of point Z corresponding to Z is obtained

z =（1+ti）/（1-ti）
=（1+ti)^2/（1+t^2）
= [1-t^2]/(1+t^2) + 2ti/(1+t^2)
The trajectory equation of point Z corresponding to Z is,
x = [1-t^2]/(1+t^2)
y = 2t/(1+t^2)
Where t is any real number

Let z = (1-radical 3I) / (radical 3 + I) ^ 2, then the absolute value of Z is?

Is it the module of Z
Z = (1-radical 3I) / [3 + (2-radical 3) I-1]
=(1-radical 3I) / [2 + (2 radical 3) I]
=(1-radical 3I) [2-2 radical 3I] / 16
=(1-radical 3I) ^ 2 / 8
=(1-2 radical 3i-3) / 8
=-(1 + radical 3I) / 4
|z|=1/2

Find the minimum value (x > 0) of y = 2x & # 178; + 3 / X and the maximum value (x ∈ R +) of y = x (1-x & # 178;)

(1) According to the mean value inequality of three variables, we get that y = 2x & # 178; + 3 / x = 2x & # 178; + 3 / (2x) + 3 / (2x) ≥ 3 · [2x & # 178; · 3 / (2x) · 3 / (2x)] ^ (1 / 3) = 3 · (9 / 2) ^ (1 / 3).. 2x & # 178; = 3 / (2x), that is, when x = (3 / 4) ^ (1 / 3), the minimum value is: 3 · (9 / 2) ^ (1 / 3). (2) according to the mean value inequality of three variables

Find the maximum m and minimum m (1) - 3 ≤ x ≤ - 2 (2) - 2 ≤ x ≤ 3 (3) 2 ≤ x ≤ 3 (3) of the function y = 1 / 2x square - x + 1 in the following range

Y = x ^ 2 / 2-x + 1 = 1 / 2 (x-1) ^ 2 + 1 / 2, the opening of the parabola is upward, the axis of symmetry is x = 1, and the vertex is (1,1 / 2). When - 3 & lt; = x & lt; = - 2, the axis of symmetry is outside this interval, and the right side is monotonically decreasing. When x = - 3, the maximum value is 17 / 2, when x = - 2, the minimum value is 5, when - 2 & lt; = x & lt; = 3