Reflections in a line (Matrices)- Please Help

[Odette]

Would anyone be able to tell me how to go about answering the following question? Thanks in advance
The point Q(5,10) is reflected in the line rx +sy= 0 and the resulting point is (198/25, 214/25). Find the values of r and s.

[enpassant]

214,25?

[Odette]

214,25?

Typing error sorry >.< lol
It's supposed to be 214/25

[enpassant]

Find the midpoint between Q and its image. Sub into the equation to find r/s.

[Odette]

Ahh and you find the midpoint how? Sorry I have forgotten a lot of things >.<
Damn i'm stupid how could i forget that >.< lol someone slap me

[Odette]

Ok so do i sub in the midpoint into rx+sy=0?

[Odette]

I'm confused

[Odette]

Hey are you sure that's what you're supposed to do? It doesn't seem right and i'm totally confused... please explain ^_^

[Odette]

Can anyone please help me? I really don't get it...

[enpassant]

midpoint = ( (5+198/25)/2, (10+214/25)/2 ) = (6.46, 9.28)

6.46r +9.28s =0, .: -r/s = 464/323

rx + sy = 0, y=(-r/s)x, y =(464/323)x, 464x - 323y = 0.

[Odette]

Ok but ah the points dont satisfy the equation... or isnt that the answer?
And how did -r/s= 464/323?

[Odette]

Please someone help me

[Odette]

Oh nevermind I figured it out... I am so stupid sometimes lol... it was so obvious >.<

[Mao]

the wording here is not exactly pedantic:

imagine your point Q being reflected by that line.
the image will be in the same position but on opposite side of that line
if you find the mid-point of those two points, you have the point where the "mirror" is, a point on your line of reflection.

so the midpoints would be:





which means:



simplifying this to:





which means, one of the possible solutions is:

and

[Odette]

Yeah I got that and a different answer... so is there more than one answer?