边长为1的正方形ABCD两顶点A,B在函数y=1／x图象上,双曲线y=k／x恰好经过正方形的另两个顶点CD,则K的?

Question

边长为1的正方形ABCD两顶点A,B在函数y=1／x图象上,双曲线y=k／x恰好经过正方形的另两个顶点CD,则K的?

Answer 1

根据对称性,正方形的两边应当与y = x平行,另两边与y = x垂直.
AB的斜率为-1,方程为y = -x + c
与y = 1/x联立,x² - cx + 1 = 0
x₁ + x₂ = c,x₁x₂ = 1
AB² = 1 = (x₁ - x₂)² + (y₁ - y₂)² = (x₁ - x₂)² + (-x₁ + c + x₂ - c)² = 2(x₁ - x₂)² = 2[(x₁ + x₂)² - 4x₁x₂]
= 2(c² - 4)
c = ±3/√2
A(1/√2,√2),B(√2,1/√2)
不妨取c = 3/√2计算(不影响结果)
AD方程为 y - √2 = x - 1/√2
与y = k/x联立,2x² + √2x -2k = 0
D([-√2 + √(2 + 16k)]/4,[√2 + √(2 + 16k)]/4)
AD² = 1
最终得16k(k - 3) = 0
k = 3