Let R be a rectangle given by the lines x=0, x=2, y=0 and y=5. Let A(α,0) and B(0,β), where α∈[0,2] and β∈[0,5], be such that the line segment AB divides the area of the rectangle R in the ratio 4:1. Then, the midpoint of AB lies on a:
Explanation
R: 0≤x≤2, 0≤y≤5
Area(R)=2×5=10
Given ratio=4:1
Smaller area=51×10=2
A(α,0), B(0,β)
Equation of AB: αx+βy=1
Area under AB=21αβ
21αβ=2
αβ=4(1)
Midpoint M=(2α,2β)
Let M(x,y)
x=2α, y=2β
xy=4αβ
Using (1),
xy=1
xy=1
Midpoint lies on a hyperbola
Explanation
R: 0≤x≤2, 0≤y≤5
Area(R)=2×5=10
Given ratio=4:1
Smaller area=51×10=2
A(α,0), B(0,β)
Equation of AB: αx+βy=1
Area under AB=21αβ
21αβ=2
αβ=4(1)
Midpoint M=(2α,2β)
Let M(x,y)
x=2α, y=2β
xy=4αβ
Using (1),
xy=1
xy=1
Midpoint lies on a hyperbola
