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.
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:
^2+(y-10)^2}=\sqrt{(x-\frac{198}{25})^2+(y-\frac{214}{25})^2})
^2+y^2-\frac{428}{25}y+(\frac{214}{25})^2)
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
 = \left(\frac{174}{25},\frac{232}{25}\right))
The rest is basically what Mao said. There are infinitely many answers, so we can take any one of them such as:
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