Coordinate geometry question, 1970's A-Level: Help!!!

[Colin67]

Hi All

First post. It's an old question from the textbook I originally used in 1980's when studying A Level maths.

For the sheer fun of it I started to study maths again.

The question is:

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

I've tried approaches using distance between points, ratio of divided line formula, finding gradient of line PQ all to no avail. I have equations with four coordinates to be found and I can't seem to eliminate them via simultaneous equations. The question is in the original Bostock and Chandler book, Pure Mathematics 1 Chapter 4. The preceding chapter covered the basics, length of line, gradient, ratio of divided line etc, nothing very advanced.

Thanks for any advice, suggestions or solutions.

 

[tkhunny]

Are you using the additional information that A, P, and Q are colinear? This provides additional information.

 

[MarkFL]

Hello, and welcome to FMH!

I'd begin by making a diagram:



Now, the family of lines through point A is:

[MATH]y=m(x-1)+2[/MATH]
To find where this line intersects the first line (point \(P\)), we may solve:

[MATH]\frac{3x-5}{2}=m(x-1)+2[/MATH]
[MATH]x=\frac{9-2m}{3-2m}\implies y=\frac{3\dfrac{9-2m}{3-2m}-5}{2}=\frac{2(3+m)}{3-2m}[/MATH]
To find where this line intersects the second line (point \(Q\)), we may solve:

[MATH]12-x=m(x-1)+2[/MATH]
[MATH]x=\frac{m+10}{m+1}\implies y=12-\frac{m+10}{m+1}=\frac{11m+2}{m+1}[/MATH]
And then we may write:

[MATH]AQ^2=\left(\frac{m+10}{m+1}-1\right)^2+\left(\frac{11m+2}{m+1}-2\right)^2[/MATH]
[MATH]AQ^2=\left(\frac{9}{m+1}\right)^2+\left(\frac{9m}{m+1}\right)^2=\frac{9^2(m^2+1)}{(m+1)^2}[/MATH]
[MATH]AP^2=\left(\frac{9-2m}{3-2m}-1\right)^2+\left(\frac{2(3+m)}{3-2m}-2\right)^2[/MATH]
[MATH]AP^2=\frac{6^2(m^2+1)}{(3-2m)^2}[/MATH]
And now, we wish to solve:

[MATH]\frac{9^2(m^2+1)}{(m+1)^2}=4\frac{6^2(m^2+1)}{(3-2m)^2}[/MATH]
[MATH]\frac{9}{(m+1)^2}=\frac{16}{(3-2m)^2}[/MATH]
[MATH]9(3-2m)^2=16(m+1)^2[/MATH]
[MATH]m=\frac{7\pm6}{2}[/MATH]
I have plotted the two solution lines as dashed lines:



Use the values of \(m\) we found to compute the two sets of points.

 

[Colin67]

Dear Mark FL

Thank you so much for help, I will work through the algebra myself from the beginning given your advice to use m as the form of the equations

 

[Colin67]

Hi

I finished the question.

Coordinates of P either (4,3.5) or (2/5, -19/10) and Q(7,5) or (2.2,9.4). Interestingly the answer in the book only gave one point for P and Q (the underlined pair). Wondered if they used a different method? Thank you again for the help, I spent a few days pondering that question!