How to find Intersection of Two Curves

Question

I need to find the intersection of curves $x^2 + y^2 + z^2 =1 $ and $x + y=0$ So using $x =-y$ . I get $2y^2 + z^2 = 1$ and this curve is an ellipse in $y-z$ plane.

However my book says the answer is a circle .

Can anyone tell me what is wrong with my solution ?

Answer 1

The equation $2y^2 + z^2 = 1$ is indeed an ellipse in the $y-z$ plane but the intersecion is in the space between a sphere and a plane through the origin which is indeed a circle.

We can use a parametric representation of the circle

• $x^2 + y^2 + z^2 =1$
• $x+y=0$

by

• $x=t$
• $y=-t$
• $z=\pm\sqrt{1-2t^2}$

with $1-2t^2 \ge 0 \implies -\frac{\sqrt 2}2\le t \le\frac{\sqrt 2}2$.