Let R be a rectangle given by the lines x=0,x=2,y=0 and y=5.

Question

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 :

Accepted Answer

We have, $R$ be a rectangle formed by the lines $x=0$, $x=2, y=0$ and $y=5$

Let $P Q R S$ be the rectangle such that

$$
P(0,0), Q(2,0), R(2,5) \text { and } S(0,5)
$$

$P Q=S R=2$ units and $P S=Q R=5$ units

$\therefore$ Area of rectangle $P Q R S=(2 \times 5) \mathrm{sq}$ units $=10 \mathrm{sq}$ units

Area of $\triangle P A B=\frac{1}{5} \times$ Area of rectangle

$$
\begin{aligned}
\frac{1}{2} \times \alpha \times \beta & =\frac{1}{5} \times 10 \text { sq units } \\
& =2 \text { sq units } \\
\frac{1}{2} \times \alpha \times \beta & =2 \\
\Rightarrow \alpha \beta & =4 \text { sq units } .........(i)
\end{aligned}
$$

Let $M(h, k)$ be the mid-point of $A B$.

Here, $h=\frac{\alpha}{2}, k=\frac{\beta}{2}$

or $\alpha=2 h$ and $\beta=2 k$

On substitute values of $\alpha$ and $\beta$ in Eq. (i), we get

$$
\begin{aligned}
\quad(2 h)(2 k) & =4 \\
\Rightarrow h k & =1 \\
\therefore \text { Locus of } M \text { is } x y & =1 \text {, which is a hyperbola. }
\end{aligned}
$$

Solutions · 1