Find a point on the line through $A=(2,3)$ and $B=(-5,-4)$ that is twice as far from $A$ as from $B$

Question

Find a point on the line through $A=(2,3)$ and $B=(-5,-4)$ that is twice as far from $A$ as from $B$. Please indicate the actual position of the point.

Answer 1

Hint:

To make things easier do it for $A'=\langle7,7\rangle=A+\langle5,4\rangle$ and $B'=\langle0,0\rangle=B+\langle5,4\rangle$.

If $P'$ is the result of this then $P=P'-\langle5,4\rangle$ works for the original question.

Answer 2

$(2,3)+(-7,-7)=(-5,-4)$. So, the step from A to B is $(-7,-7)$. However, you want to be twice as far from A as from B. So you want to go $\frac{2}{3}$ of the way. So instead of taking the $(-7,-7)$ step from A, you'll take a $\frac{2}{3}(-7,-7)=(\frac{-14}{3},\frac{-14}{3})$ step. Hence you'll end up at $(2,3)+(\frac{-14}{3},\frac{-14}{3})=(2-\frac{14}{3},3-\frac{14}{3})$.

Answer 3

There are two solutions. For instance, if $A$ were $(0,0)$ and $B$ were $(1,0)$, then the two solutions would be $(\frac23,0)$ and $(2,0)$. Can you use this idea to solve your actual problem?

Answer 4

The question in OP is a particular case of the more general problem:

Given two points $A$, $B$, find a point $X$ on the straight line passing through these points such that $n\overline{AX}=m\overline{BX}$.

This problem has two solution: one for $X$ internal to the segment $\overline{AB}$ and the other for $X$ external to the segment.

The solution is easily found using vectors as you can see in the figures (here the vectors are different than in OP for better readability) .

If $ X$ is inside we have:

$$ n(\vec w-\vec x)=-m(\vec v - \vec x) $$ with a minus sign of RHS because the two vectors $(\vec w-\vec x)$ and $(\vec v - \vec x)$ are opposite. And, solving for $\vec x$ we find: $$ \vec x= \dfrac{n\vec w +m\vec v}{m+n} $$

If $ X$ is outside we have: $$ n(\vec w-\vec x)=m(\vec v - \vec x) $$ where LHS and RHS have the same sign because the two vectors have the same orientation, and we find: $$ \vec x= \dfrac{n\vec w -m\vec v}{n-m} $$ These are the ''section formula'' cited by @yashg in his comment. Substitung the coordinate of the given points and $n=1$, $m=2$ you can solve the question.