Question

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), 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 mid-point of AB lies on a 

(1) parabola 

(2) hyberbola 

(3) straight line 

(4) circle

Answer 1

(2) hyberbola 

\(\frac{ar(OPQR)}{or(OAB)}=\frac{4}{1}\)

Let M be the mid-point of AB.

∴ Locus of M is xy = 1 

Which is a hyperbola.