A straight fence runs from post \(A(-3,\,5)\) to post \(B(6,\,-1)\). Two more posts, P and Q, are fixed on the fence so that \(AP=PQ=QB\), with P nearer to A. Where is post Q?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((3,\,1)\)
\(AP=PQ=QB\) means P and Q cut AB into three equal parts, so P divides AB in the ratio \(1:2\) and Q divides AB in the ratio \(2:1\) (two parts from A, one part from B). The section formula gives Q.
For Q: \(m=2\), \(n=1\), \(A(-3,\,5)\), \(B(6,\,-1)\)
x-coordinate: \(\displaystyle \frac{2\times6+1\times(-3)}{2+1}=\frac{12-3}{3}=3\)
y-coordinate: \(\displaystyle \frac{2\times(-1)+1\times5}{2+1}=\frac{-2+5}{3}=1\)
So \(Q=(3,\,1)\)
Check: P (ratio \(1:2\)) is \((0,\,3)\); the steps A to P, P to Q and Q to B are each \((+3,\,-2)\), so the three parts are equal
