Given * * a = {(x, y) | ax + y = 1}, B = {(x, y) | x + ay = 1}, C = {(x, y) | X & # 178; + Y & # 178; = 1} then (1) What is the value of a when (a ∪ b) ∩ C is a 2-ary set (2) What is the value of a when (a ∪ b) ∩ C is a 3-ary set The answer is (1) a = 0 or 1 (2) a = - 1 ± √ 2, And I'm just a freshman,

Given * * a = {(x, y) | ax + y = 1}, B = {(x, y) | x + ay = 1}, C = {(x, y) | X & # 178; + Y & # 178; = 1} then (1) What is the value of a when (a ∪ b) ∩ C is a 2-ary set (2) What is the value of a when (a ∪ b) ∩ C is a 3-ary set The answer is (1) a = 0 or 1 (2) a = - 1 ± √ 2, And I'm just a freshman,

(1) Analysis: with the combination of number and shape, a and B represent the straight line of constant crossing point (0,1) and constant crossing point (1,0) respectively, while C represents a unit circle with the center of a circle at the origin. (a ∪ b) ∩ C is a set of 2 elements, indicating that the straight line represented by a and B has and only has two intersections with the unit circle respectively

x. Y is a positive real number, and x ^ 2 + y ^ 2 / 2 = 1, find the maximum value of X * √ 1 + y ^ 2

X ^ 2 + y ^ 2 / 2 = 1
2x^2+(1+y^2)=3
2x^2+(1+y^2)≥2√(2x^2*(1+y^2))
3≥2√2*x*√(1+y^2)
x*√(1+y^2)≤3√2/4
If and only if x = √ 3 / 2, y = ± √ 2 / 2, the maximum value of X * √ 1 + y ^ 2 is 3 √ 2 / 4

What is the maximum value of YX in all real number pairs (x, y) satisfying (x-3) 2 + (Y-3) 2 = 6?

Let y = KX, then when the line y = KX is tangent to the circle (x-3) 2 + (Y-3) 2 = 6, K has the maximum and minimum value. Substituting y = KX into (x-3) 2 + (Y-3) 2 = 6, we can get (1 + K2) x2-6 (K + 1) x + 12 = 0, that is, k2-6k + 1 = 0. Solving this equation, we can get k = 3 + 22 or 3-22. The maximum value of YX = k is 3 + 22