1. plane geometry

Question: A line is drawn through the point A (1,2) to the cut the line $\displaystyle 2y=3x-5$ in P and the line $\displaystyle x + y = 12$ in Q. If AQ = 2AP, find the coordinates of P and Q.

I have drawn a sketch but desperately need help i cant figure out how to answer it

2. Re: plane geometry

For me, this was quite a problem. There's probably an easier way to solve this but the following sledge "hammer approach" works. I needed the computer to help me solve an equation.

Here's a drawing of the situation with m=1/2:

3. Re: plane geometry

Let
$\displaystyle P=(a,b)$ and $\displaystyle Q=(c,d)$

$\displaystyle \overrightarrow{\text{AQ}}=\pm 2 \overrightarrow{\text{AP}}$

Translating into coordinates gives 2 linear equations

$\displaystyle \pm 2(a-1)=c-1$

$\displaystyle \pm 2(b-2)=d-2$

Two more linear equations:

$\displaystyle 2b=3a-5$

$\displaystyle c+d=12$

EDIT: Looking at the sketch provided by the OP, APQ doesn't look at all like a straight line in which case the problem has infinitely many solutions. Otherwise, only two solutions

4. Re: plane geometry

As usual, Idea gave a much nicer solution than mine. However, by hand his solution still requires some grunt work. Here are some observations:

5. Re: plane geometry

Originally Posted by Idea
Let
$\displaystyle P=(a,b)$ and $\displaystyle Q=(c,d)$

$\displaystyle \overrightarrow{\text{AQ}}=\pm 2 \overrightarrow{\text{AP}}$

Translating into coordinates gives 2 linear equations

$\displaystyle \pm 2(a-1)=c-1$

$\displaystyle \pm 2(b-2)=d-2$

Two more linear equations:

$\displaystyle 2b=3a-5$

$\displaystyle c+d=12$

EDIT: Looking at the sketch provided by the OP, APQ doesn't look at all like a straight line in which case the problem has infinitely many solutions. Otherwise, only two solutions
where is the solution did you edit it

6. Re: plane geometry

Originally Posted by johng
As usual, Idea gave a much nicer solution than mine. However, by hand his solution still requires some grunt work. Here are some observations:

But can I ask how do you do the matrix as normal simultaneous equation as I have not yet being taught matrix

7. Re: plane geometry

Originally Posted by johng
For me, this was quite a problem. There's probably an easier way to solve this but the following sledge "hammer approach" works. I needed the computer to help me solve an equation.

Here's a drawing of the situation with m=1/2:

I understand but I want to ask just one thing
how do you get for line l: y=m(x-1) +2 where did it come from

thank you