Find the coordinates of P and Q.

Question

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

From: Mathematics, The Core Course for A-level, Bostock and Chandler. Chapter 4 Q15.

The answer is $(4,3.5)(7,5)$or$(0.4,-1.9)(2.2,9.8)$

I have no idea where to begin with this question.

I've tried letting $P=(x,12-x)$ and $Q=(x,\frac{3}{2}x-\frac{5}{2})$ and then used the distance formula to try and equate $|AQ|=2|AP|$ but didn't get anywhere.

How should I approach this problem?

Yes , I drew a diagram and the points given by the book make perfect sense. But It didn't help me to figure out how to go about this problem. — Kantura, Jul 15 '16 at 15:35

score 1 · Accepted Answer · answered Jul 15 '16 at 15:39

We can set $P(p,\frac{3p-5}{2}),Q(q,12-q)$. (you have errors here.)

From $AQ=2AP$, we have the following two cases :

Case 1 : $P$ is the midpoint of the line segment $AQ$. $$p=\frac{1+q}{2}\quad \text{and}\quad \frac{3p-5}{2}=\frac{2+(12-q)}{2}$$ Solving the system gives $p=4, q=7$.

Case 2 : $A$ is on the line segment $PQ$ where $AQ:AP=2:1$. $$1=\frac{2\times p+1\times q}{1+2}\quad \text{and}\quad 2=\frac{1\times (12-q)+2\times \frac{3p-5}{2}}{1+2}$$ Solving the system gives $p=0.4,q=2.2$.

Great, I understand now. Thanks mathlove. – Kantura Jul 15 '16 at 15:51 — Kantura, Jul 15 '16 at 15:51

Find the coordinates of P and Q.

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