How do you convert the parametric equations into a Cartesian equation by eliminating the parameter r: x=(r^2)+r, y=(r^2)-r?
2 Answers
x^2+y^2 -2x-2y -2xy = 0
Explanation:
We have:
x=r^2 + r
y=r^2 - r
Adding the equations:
x+ y = 2r^2 => r^2 = 1/2(x+y)
Multiplying the Equations we get:
xy = (r^2 + r)(r^2 - r)
\ \ \ \ = r^4 - r^2
And substituting
xy = (1/2(x+y))^2 - 1/2(x+y)
Thus the Cartesian equation is:
xy = 1/4(x+y)^2 - 1/2(x+y)
4xy = (x^2+2xy+y^2) -2(x+y)
4xy = x^2+2xy+y^2 -2x-2y
x^2+y^2 -2x-2y -2xy = 0
Explanation:
We have:
Summing the equations we have:
and subtracting the second from the first:
or:
Then:
and finally:
This is the equation of a conic, so we can calculate the invariants:
The cubic invariant is non null so the conic is non-degenerate,
the quadratic invariant is null so the conic is a parabola.
graph{x^2+y^2-2xy-2x-2y=0 [-213.9, 213.8, -106.2, 107.5]}