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  • Let R be a rectangle given by the line x = 0, x = 2, y = 0 and y = 5. Let A (, 0) and B (0, <img alt="\beta" src="

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

\beta \epsilon [0, 5], be such that the line segment AB divides the area of the rectangle R in the ratio 4 : l. Then, the mid-
point of AB lies on a :

Option: 1

straight line


Option: 2

parabola


Option: 3

circle


Option: 4

hyperbola


Answers (1)

best_answer

\frac{\operatorname{ar}(\mathrm{OPQR})}{\operatorname{or}(\mathrm{OAB})}=\frac{4}{1} Let \mathrm{M} be the mid-point of \mathrm{AB}

\begin{aligned} & \mathrm{M}(\mathrm{h}, \mathrm{k}) \equiv\left(\frac{\alpha}{2}, \frac{\beta}{2}\right) \\ & \Rightarrow \frac{10-\frac{1}{2} \alpha \beta}{\frac{1}{2} \alpha \beta}=4 \\ & \Rightarrow \frac{5}{2} \alpha \beta=10 \Rightarrow \alpha \beta=4 \\ & \Rightarrow(2 \mathrm{~h})(2 \mathrm{~K})=4 \\ & \therefore \text { Locus of } \mathrm{M} \text { is } \mathrm{xy}=1 \\ & \text { Which is a hyperbola. } \end{aligned}

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Ritika Harsh

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