# Thread: [SOLVED] Analytic Geometry Q9

1. ## [SOLVED] Analytic Geometry Q9

Question:
A line passes through the points: $A(1,1)$ & $B(5,7)$. Find the coordinates of $P$ if $P$ is three times as far from $A$ as from $B$.

Attempt:
I know that I can find the points of $P$ using this formula: $X = x_1 + r(x_2 - x_1$) & $Y = y_1 + r(x_2 - x_1)$, but how can I find $r$?

2. Originally Posted by looi76
Question:
A line passes through the points: $A(1,1)$ & $B(5,7)$. Find the coordinates of $P$ if $P$ is three times as far from $A$ as from $B$.

Attempt:
I know that I can find the points of $P$ using this formula: $X = x_1 + r(x_2 - x_1$) & $Y = y_1 + r(x_2 - x_1)$, but how can I find $r$?
There are two different points P. In my sketch one point is $P_{red}$ and the other is $P_{blue}$.

1. $P_{red}$: Use proportions:

$\dfrac{AP}{PB}=\dfrac31~\implies~\dfrac{AP}{AB-AP}=3$

Solve for AP. I've got $AP=\underbrace{\frac34}_{\text{= r}} \cdot AB$

2. If you use the equation of the line:

$l:\left\{\begin{array}{l}x=1+r\cdot (5-1) \\ y=1+r \cdot (7-1)\end{array}\right.~\wedge r=\frac34~\implies~ \begin{array}{l}x=4 \\ y=\frac{11}2\end{array}$

3. I leave the calculation of the coordinates of $P_{blue}$ to you. (For confirmation only: (7, 10))