A straight pipeline runs from a tank at \(A(-2,\,1)\) to a pump at \(B(8,\,6)\). A valve \(V\) is fitted on the pipeline so that \(AV:VB=2:3\). Where is the valve?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((2,\,3)\)
By the section formula, the point dividing the segment from \(A(x_1,\,y_1)\) to \(B(x_2,\,y_2)\) internally in the ratio \(m:n\) is \(\left(\frac{mx_2+nx_1}{m+n},\,\frac{my_2+ny_1}{m+n}\right)\): each end is weighted by the part of the ratio next to the other end.
Here \(m=2\), \(n=3\), \(A(-2,\,1)\) and \(B(8,\,6)\)
x-coordinate \(\displaystyle =\frac{2(8)+3(-2)}{2+3}=\frac{16-6}{5}=2\)
y-coordinate \(\displaystyle =\frac{2(6)+3(1)}{2+3}=\frac{12+3}{5}=3\)
So the valve is at \(V(2,\,3)\)
