Question

Consider the implicitly defined surface 2xyz +xy +z 2 + 2 = xz2 +x+y + 2z in R 3 . (a) Find the two points of intersection of the surface with the line through the origin and in direction j + k.

Answer 1

The two points of intersection are:

(0, 1, 1) and (0, 2, 2)

Step-by-step explanation:

The parametric equations of the line passing through the origin (0, 0, 0) and in direction j + k is:

`x=0+0t=0` Notice x is 0t because vector j+k has no x-component

`y=0+1t=t`

`z=0+1t=t`

Then we plug them into the equation of the surface:

`2xyz+xy+z^2+2=xz^2+x+y+2z`

And we get:

`2(0)(t)(t)+(0)(t)+t^2+2=(0)t^2+0+t+2t`

`t^2+2=t+2t`

Collecting all term on the left side:

`t^2-3t+2= 0`

Factoring:

`(t-2)(t-1)=0`

Solving each factor:

`t=2, t=1`

Then plugging the solution into the parametric equations of the line we get:

`\text{For }t=2\to x=0,y=2,z=2\\\text{For }t=1\to x=0,y=1,z=1`

So, the points of intersection are: (0, 1, 1) and (0, 2, 2)