Expert: Steve Holleran - 10/7/2007

Question
QUESTION: if p(t + 1/t, t - 1/t) is a variable point, find it's locus is the question.

what is the method to find the locus of a point, where the co-ordinates of only a single point are given?

ANSWER: Hi Aarthi,

Not much information here, but my guess is that you just allow t to take on integer values, ...-3,-2, -1, , 1, 2 ,... and get the x and y values:

  t          p

 1          (2,0)

 2          (3/2, 1/2)

 3          (4/3, 2/3)

 -1          (0,2)

etc.

That's the only thing I can think of .
Steve

---------- FOLLOW-UP ----------

QUESTION: actually, the answer is given as x^2 - y^2 = 4,
i just can't understand how this was arrived at.  

Answer
Hi Aarthi,

Well, like a lot of things, now that I see the answer, I think this is what they did:

x = t + 1/t  so x^2 = t^2 + 2 + 1/t^2

y = t - 1/t, so y^2 = t^2 - 2 + 1/t^2

then x^2 - y^2 = (t^2 + 2 + 1/t^2) - (t^2 - 2 + 1/t^2)

and after you distribute the - sign on the second parenthesis, you have x^2 - y^2 = 4.

What I don't get is how anyone would know to do that.
Its been awhile since I've dealt with locus problems, so maybe there are some set strategies that are used that I've forgotten.

Thanks for your update, aarthi

Steve