this is not a homework question and I consider this problem quite difficult and confusing. I tried hard to solve it for 2 days, sure I found solutions, but they are not the same that the one provided by the book. I even know the problem by heart, I don't even need to check the book to write it down. The problem is
Transform this parametric equation in rectangular form:
$x = a\frac{2t}{1+t^2}$
$y = a\frac{1-t^2}{1+t^2}$
The solution provided by the book is
$x^2 + y^2 = a^2$
So I know that this is a circle where a is the radius. It's a circle that moves around the cartesian plane, diagonally, describing a negative slope. I also know that t is a point in the circonference and finally x and y express the position of the circle in the plane.
But the problem is that I found no way to eliminate the parameter. I even tried with a radius a=1 and this doesn't lead me to the desired solution. I always get the parameter into the equation and never manage to eliminate it. I tried similar problems, looked for solutions with online calculators. I even got
$|t| = \frac{\sqrt{(-x+2at)x}}{x}$
or this
$t=\frac{2-\sqrt{4-4x^2}}{2x}$
for a=1
But these equations lead to nothing. I tried partial fractions, polar coordinates, algebraic transformations, nothing worked.
I know that it's ok if we, as a students, can't solve some problems, I also know that if I can't solve a problem in 2 days, I would better to drop it and go to the next. But the book solution makes sense and it would be instructive to see how to reach it.
Best Answer
This should be a very easy problem. Try to calculate $x^{2}+y^{2}$ and simplify. See what you get in terms of a.