Question

如图,直线Y=X+3与x轴,y轴分别交于A,B点,与Y=K/X(X<0） 的图象交于C,D点.E是点C关于点A的中心对称点

EF⊥OA于F。若△AOD的面积与△AEF的面积之 和为
7/2时，则k=----------

Answer 1

A(-3, 0), B(0, 3)
y = x + 3 = k/x
x² + 3x - k = 0
x₁ = [-3 - √(9 + 4k)]/2, x₂ = [-3 + √(9 + 4k)]/2
按图C(x₁, x₁ + 3), D(x₂, x₂ + 3)
E(e, e')
(e + x₁)/2 = -3, e = -6 - x₁
(e' + x₁ + 3)/2 = 0, e' = -x₁ - 3
E(-6 - x₁, -x₁ - 3)

F(-6-x₁, 0)
△AOD的面积 + △AEF的面积
= (1/2)AO*D的纵坐标 + (1/2)FA*|E的纵坐标|
= (1/2)*3*(x₂ + 3) + (1/2)(-3 + 6 + x₁)|-x₁ - 3|
= (3/2)(x₂ + 3) + (1/2)(x₁ + 3)²
= 9/4 + (3/4)√(9 + 4k) + 9/4 + 4k -6√(9 + 4k)
= 9/2 + k/2 = 7/2

k = -1