Let R be the rectangle given by the lines x = 0 , x = 2 , y = 0 and y = 5 . Let A ( 0 ) and B ( 0 , ) , [ 0 , 2 ] and [ 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 l

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Solution

Q.
Let R be the 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 midpoint of AB lies on a [2023]

1 parabola

2 hyperbola

3 straight line

4 circle

Ans.
(2)

$\frac{ar (A P Q R B)}{ar (∆ O A B)} = \frac{4}{1}$

Let M be the mid-point of AB.

$M (h, k) \equiv (\frac{α}{2}, \frac{β}{2})$

$\Rightarrow \frac{10 - \frac{1}{2} α β}{\frac{1}{2} α β} = 4 \Rightarrow \frac{5}{2} α β = 10$

$\Rightarrow α β = 4$

$\Rightarrow (2 h) (2 k) = 4$

$∴$ $Locus of M is x y = 1, which is a hyperbola.$

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