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Reflections in a line (Matrices)- Please Help

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Mao:
looking at

its a relationship, that there are an infinite number of answers.

to express one in terms of another:



so whatever values you plug into s (anything, like 1 or , or to be more pedantic, ), you'll have an r counterpart.

however, the values of r and s doesnt really affect the line (so long as they follow the above relation), as it is the same way of representing the line:







the ratio of r/s will always stay the same, but there are infinite number of ways you can represent this ratio.

Odette:
Thanks Mao I get it now :)

Neobeo:

--- Quote from: Odette on April 04, 2008, 08:45:24 am ---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.
--- End quote ---

I'm not sure if it was a typo or a question intentionally made to be different, but it does not have a solution. Let M be the midpoint of P(198/25, 214/25) and Q(5,10). From the question statement, Q is reflected in OM onto the point P, implying that OM and PQ are perpendicular. This further implies that OPQ is an isosceles triangle with OP = OQ. However a quick check shows that:




So there are no solutions to this question.

Incidentally, a line that reflects point Q to P does exist; it just does not pass through the origin as implied by the question. If we do want to find this line, we could just find the locus of all points that are equidistant from P and Q, i.e:




The and terms conveniently cancel out, simplifying it to:



Sure enough, this cannot be made into the form rs + xy = 0.


If, however, we change the question slightly to let Q be the coordinate (6,10), then there will be a solution. A quick check shows that .

By this new question, then we can do what Mao and enpassant suggested, i.e.

Midpoint



The rest is basically what Mao said. There are infinitely many answers, so we can take any one of them such as:

Odette:
Ok you've lost me lol... Ah it's not a typo though that's the question that was given...
So there's no solution? I'm confused now :(

Neobeo:

--- Quote from: Odette on April 05, 2008, 10:33:49 am ---Ok you've lost me lol... Ah it's not a typo though that's the question that was given...
So there's no solution? I'm confused now :(

--- End quote ---

Yup, I'm saying there is no solution. Here is an exaggerated diagram to explain why:



We plot P and Q on the graph, and find the midpoint M. Since the line is of the the form rs + xy = 0, it must pass through the origin. So we have a unique line defined by extending OM. However, if we reflect Q in the line OM, we see that it ends up at Q' (green point) rather than onto P.

If, however, the question stated that P was actually at Q', then all is good and well. See how Q and Q' are at equal distances from the origin.

If you can let the line not pass through the origin though, then it would be the line showed by the dark blue dotted lines.

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