1. ## Linear functions

If A = (-4, 6) and B = (6, -7) find:
I) the coordinates of P, where P € AB and AP:PB = 3:1
ii) the coordinates of P, where P € AB and AP:AB = 3:1

2. $\displaystyle P \in AB, AP:PB=3:1$
p is in AB, The ratio of AP to PB is 3 to 1.... (right?)
$\displaystyle a=(-4,6) b=(6,-7)$
I.
$\displaystyle y=-1.3x+0.8$
AB:
$\displaystyle \Delta y=13$
$\displaystyle \Delta x=10$

$\displaystyle AB=\sqrt{a^{2}+b^{2}}$
$\displaystyle AB=\sqrt{\Delta x^{2}+\Delta y^{2}}$
$\displaystyle AB=\sqrt{13^{2}+10^{2}}$
$\displaystyle AB=\sqrt{269}$

AP:

$\displaystyle \frac{3\sqrt{269}}{4}$

PB:

$\displaystyle \frac{\sqrt{269}}{4}$

and just use $\displaystyle \Delta x$ and $\displaystyle \Delta y$

to figure out the coordinates.
If this is correct just do the same for II,

3. Hello scubasteve94
Originally Posted by scubasteve94
If A = (-4, 6) and B = (6, -7) find:
I) the coordinates of P, where P € AB and AP:PB = 3:1
ii) the coordinates of P, where P € AB and AP:AB = 3:1
I'm sure you know the formula for finding the coordinates of the mid-point of the line joining $\displaystyle A\;(x_1,y_1)$ to $\displaystyle B\;(x_2,y_2)$. It is:
$\displaystyle \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Well, there's a more general formula for the coordinates of the point $\displaystyle P$ that divides $\displaystyle AB$ in the ratio $\displaystyle m:n$. It is this:
$\displaystyle \left(\frac{nx_1+mx_2}{m+n},\frac{ny_1+my_2}{m+n}\ right)$
(If you understand vector notation, this is equivalent to $\displaystyle \textbf{p}=\frac{n\textbf{a}+m\textbf{b}}{m+n}$, where $\displaystyle \textbf{a},\textbf{b},\textbf{p}$ are the position vectors of $\displaystyle A, B$ and $\displaystyle P$.)

For part (i), $\displaystyle AP:PB = 3:1$, so $\displaystyle m=3, n = 1$. So plug these values into the formula, using $\displaystyle x_1=-4,y_1=6,x_2=6,y_2=-7$.

For part (ii), we have $\displaystyle AP:AB=3:1$. So $\displaystyle P$ lies outside the line segment $\displaystyle AB$. Draw a diagram if you're not sure. You'll then find that the ratio $\displaystyle AP:PB = 3:-2$. So you'll need to use the values $\displaystyle m = 3, n = -2$.

(i) $\displaystyle (\tfrac72,-\tfrac{15}{4})$; (ii) $\displaystyle (26,-33)$