Intersecting lines; distance ratios

**Harry1W** · September 28th 2009, 02:46 PM

I'm really stumped on this question, but I'm sure the answer is really simple!

"There exist two lines $2y = 3x - 5$ and $x+y=12$ . A line passing through a pt A $(1, 2)$ intersects these lines at the pts P and Q, respectively. The distance AQ is twice that of AP. What are the co-ordinates of P and Q?"

So far, I have said:

$<br /> AQ = 2 \cdot AP$ .

So,

$AQ^2 = 4AP^2$ .

If we denote the coordinates of P as $(a, b)$ and those of Q, $(c, d)$ , then by Pythagoras' Theorem,

$AQ^2 = (c - 1)^2 + (d - 2)^2$

and $AP^2 = (a - 1)^2 + (b - 1)^2$ .

We can therefore substitute these two equations into $AQ^2 = 4AP^2$ to give an equation in $a, b, c$ and $d$ . Then, using the equations of the lines we can see that we can rewrite $b$ as

$b = \frac{3a - 5}{2}$

and $d$ as

$d = 12 - c$

which we can then use to derive an equation in terms of $a$ and $c$ . But how to eliminate one of these variables eludes me! Your help is much appreciated.