Co-ordinate Geometry + Division of Interval

[DanielSmith]

3U mathematics

Thanks

[RuiAce]

\[ \text{The point }P\text{ can be parametrised as }\left(t, \frac{3t-5}{2} \right).\\ \text{The point }Q\text{ can be parametrised as }\left( s, 12-s\right). \]
\[ \text{Once we have this, we can just plug straight into the ratio division formula.}\\ \text{Considering }AP:AQ = 1:2\text{ we have}\\ \begin{align*} 1 &= \frac{s+2t}{1+2}\\ 2&= \frac{(12-s) + 2\left(\frac{3t-5}{2}\right)}{1+2} \end{align*}\\ \text{You should now be able to solve these simultaneous equations.}\]
Note: Here I assume internal division, although in theory external division is also possible.

[DanielSmith]

\[ \text{The point }P\text{ can be parametrised as }\left(t, \frac{3t-5}{2} \right).\\ \text{The point }Q\text{ can be parametrised as }\left( s, 12-s\right). \]
\[ \text{Once we have this, we can just plug straight into the ratio division formula.}\\ \text{Considering }AP:AQ = 1:2\text{ we have}\\ \begin{align*} 1 &= \frac{s+2t}{1+2}\\ 2&= \frac{(12-s) + 2\left(\frac{3t-5}{2}\right)}{1+2} \end{align*}\\ \text{You should now be able to solve these simultaneous equations.}\]
Note: Here I assume internal division, although in theory external division is also possible.

Thank you for replying.
I was able to get the right answer using your method. Q(11/5, 49/5) internally, Q(7,5) externally.