Let R be a rectangle given by the lines $x=0, x=2$, $y=0$ and $y=5$. Let $A(\alpha, 0)$ and $B(0, \beta), \alpha \in[0,2]$ and $\beta \in[0,5]$, 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

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Straight Lines

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 \in [0, 2]

and

β \in [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

(A)parabola
(B)hyperbola✓ Correct
(C)straight line
(D)circle

Answer: (B)

Solution

The rectangle has area 10. The line AB divides it in ratio 4:1, so the triangle OAB has area 2 (since the larger region has area 8). The area of triangle OAB is

\frac{1}{2} α β = 2

, giving

α β = 4

. Let the midpoint M be

(h, k) = (α /2, β /2)

. Then

α = 2 h

and

β = 2 k

, so

(2 h) (2 k) = 4

, which gives

hk = 1

. The locus is

x y = 1

, which is a hyperbola.

JEE Main · 2023 · 11 Apr 2023 · Shift 1 · Single correct option

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