Straight Lines
Variable Line — Minimum Sum of Intercepts
nta_pyq_2024_apr
Grade 11
Question:
Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is
Step-by-Step Solution
Key Concept: Line: $y+9=m(x-4)\Rightarrow A=\left(\frac{9+4m}{m},0\right)$, $B=(0,-9-4m)$... wait: $y$-intercept when $x=0$: $y=-9-4m$... but $m>0$ gives $B=(0,-(9+4m))$. $OA+OB=\frac{9}{m}+4m+9+4m=13+\frac{9}{m}+4m$.
By AM-GM on $4m+9/m\geq12$, minimum $OA+OB=25$.
Correct Answer: 2