Straight Lines
Line Dividing Rectangle — Locus of Midpoint
nta_pyq_2023_apr
Grade 11
Question:
Let $R$ be rectangle $x=0,x=2,y=0,y=5$. $A(\alpha,0)$, $B(0,\beta)$ divide rectangle area in ratio $4:1$. Midpoint of $AB$ lies on a
straight line
parabola
hyperbola
circle
Step-by-Step Solution
Key Concept: Area of triangle $OAB=\frac{\alpha\beta}{2}$. Ratio $4:1\Rightarrow\frac{\alpha\beta}{2}:(10-\frac{\alpha\beta}{2})=4:1\Rightarrow\alpha\beta=8$. Midpoint $(\frac{\alpha}{2},\frac{\beta}{2})$: $\frac{\alpha}{2}\cdot\frac{\beta}{2}=2$, i.e., $xy=2$ — hyperbola.
Midpoint locus: $xy=2$ (hyperbola).
Correct Answer: 3