Answer: The correct option is
(A) (11, 3).
Step-by-step explanation: We are given to find the co-ordinates of he point that is one-sixth of the way from A(14, −1) to B(−4, 23).
As shown in the attached figure below, let point P is one-sixth of the way from A to B.
Also, let P divides the the segment AB in the ratio m : n.
Then, we must have
[tex]\dfrac{m}{m+n}=\dfrac{1}{6}\\\\\\\Rightarrow 6m=m+n\\\\\Rightarrow 6m-m=n\\\\\Rightarrow 5m=n\\\\\Rightarrow \dfrac{m}{n}=\dfrac{1}{5}\\\\\Rightarrow m:n=1:5.[/tex]
So, the point P divides AB in the ratio 1 : 5.
We know that
if a point divides the line segment joining the points (a, b) and (c, d) in the ratio m : n, then its co-ordinates are
[tex]\left(\dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n}\right).[/tex]
Therefore, the co-ordinates of the point P will be
[tex]\left(\dfrac{1\times(-4)+5\times14}{1+5},\dfrac{1\times23+5\times(-1)}{1+5}\right)\\\\\\=\left(\dfrac{-4+70}{6},\dfrac{23-5}{6}\right)\\\\\\=\left(\dfrac{66}{6},\dfrac{18}{6}\right)\\\\=(11,3).[/tex]
Thus, the required co-ordinates of the point is (11, 3).
Option (A) is CORRECT.
