A straight cable is stretched from the top of a pole at \(A(-6,\,7)\) to the top of another pole at \(B(9,\,-3)\), drawn on a grid. A bird sits on the cable at the point \(P\) for which \(AP:PB=3:2\). Where is the bird?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \((3,\,1)\)
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 \(m=3\), \(n=2\), \(A(-6,\,7)\) and \(B(9,\,-3)\)
x-coordinate \(\displaystyle =\frac{3(9)+2(-6)}{3+2}=\frac{27-12}{5}=3\)
y-coordinate \(\displaystyle =\frac{3(-3)+2(7)}{3+2}=\frac{-9+14}{5}=1\)
So the bird is at \(P(3,\,1)\)
