Answer:
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:
[tex] x=0+0t=0 [/tex] Notice x is 0t because vector j+k has no x-component
[tex] y=0+1t=t [/tex]
[tex]z=0+1t=t[/tex]
Then we plug them into the equation of the surface:
[tex]2xyz+xy+z^2+2=xz^2+x+y+2z[/tex]
And we get:
[tex]2(0)(t)(t)+(0)(t)+t^2+2=(0)t^2+0+t+2t[/tex]
[tex]t^2+2=t+2t[/tex]
Collecting all term on the left side:
[tex] t^2-3t+2= 0[/tex]
Factoring:
[tex](t-2)(t-1)=0[/tex]
Solving each factor:
[tex]t=2, t=1[/tex]
Then plugging the solution into the parametric equations of the line we get:
[tex]\text{For }t=2\to x=0,y=2,z=2\\\text{For }t=1\to x=0,y=1,z=1[/tex]
So, the points of intersection are: (0, 1, 1) and (0, 2, 2)
