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yy205001
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Homework Statement
Show that the circle that is the intersection of the plane x + y + z = 0 and the
sphere x2 + y2 + z2 = 1 can be expressed as:
x(t) = [cos(t)-sqrt(3)sin(t)]/sqrt(6)
y(t) = [cos(t)+sqrt(3)sin(t)]/sqrt(6)
z(t) = -[2cos(t)]/sqrt(6)
Homework Equations
The Attempt at a Solution
At first, I tried to let z=-x-y then sub it into the equation of the sphere:
x2+y2+(-x-y)2=1
x2+y2+x2+2xy+y2=1
2x2+2xy+2y2 = 1
And i stuck at here, I can't make it into a circle equation.
So, I changed another way.
I equated two equations, so it becomes:
x2 + y2 + z2 -1 = x+y+z
x2+ x + y2 + y + z2 +Z -1 =0
(x-1/2)2 + (y-1/2)2 + (z-1/2)2 = 7/4 *By completing the square
But isn't this is an equation of a sphere??
Shouldn't it just a circle?
Any help is appreciated!