Question

Parameterize the curve of intersection of the surfaces so that the direction is clockwise when viewed from above. Include a sketch of each surface and the curve. x + z = 1 and 2 = 4 - x2 - y2

Answer 1

Answer is attached in images

Explanation:

Given:

x+z=1 and

`z=4-x^(2) -y^(2)`

Now,

`1-x=4-x^(2) 2-y^(2) 2\\x^(2) -x+y^(2) \\x^(2) -x+1/4+y^(2) =3\\(x-1/2)^(2) +y^(2) =3+1/4\\(x-1/2)^(2) +y^(2) =(\sqrt(13)/2)^(2)`

Which is circle with center (1/2,0) radius

`√(13/2)` take,
`x=1/2+√(3) /2 cos\alpha \\y=√(13)/2 sin\alpha 0\leq \alpha \leq 2\pi`

Now,