Respuesta :
Answer:
If p and q lie on the surface , then we should be able to find values of u and v such that they can satisfy the equation of the surface.
i.e
x = u+v;
y = [tex]u^2 - v[/tex] and
z = [tex]u + v^2[/tex]
For p we have to find u and v such that:
u + v =3 ; ......[1]
[tex]u^2 - v[/tex] = 3 ; .....[2]
[tex]u + v^2[/tex] = 3 .....[3]
Equate [1] and [3] we have;
u + v = [tex]u + v^2[/tex]
Subtract u from both sides we get;
u + v -u= [tex]u + v^2-u[/tex]
Simplify:
[tex]v = v^2[/tex]
or
[tex]v^2 - v = v(v-1) = 0[/tex]
⇒ v = 0 and v = 1
To find the value of u substitute the values of v in [1]
for v =0
u + 0 = 3
u = 3
for v = 1
u + 1 = 3
Subtract 3 from both sides we get;
u = 2
Now; substitute the values of u and v in equation [2] to satisfy:
if u = 3 ad v = 0
then;
[tex]u^2 - v[/tex] = 3
[tex]3^2 - 0[/tex] = 3
9 = 3 which is not true
if u =2 and v =1
then;
[tex]2^2 - 1[/tex] = 3
4-1 = 3
3 = 3 which is true.
Therefore, the point p lies on the given surface with u = 2 and v =1
Similarly, for q we have to find u and v such that:
u + v =4 ; ......[1]
[tex]u^2 - v[/tex] = -2 ; .....[2]
[tex]u + v^2[/tex] = 10 .....[3]
Adding first two equation; we get
[tex]u + v +u^2 -v = 4-2[/tex]
Simplify:
[tex]u^2+u -2 =0[/tex]
or
[tex](u+2)(u-1)[/tex] = 0
⇒ u = -2 and u = 1
from [1] we have v = 4- u
For u = -2
then;
v = 4-(-2) = 4+2
v = 6
For u = 1
then;
v = 4-1
v =3
Now; substitute the values of u and v in equation [3] to satisfy:
if u = -2 ad v = 6
then;
[tex]u +v^2[/tex] = 10
[tex]-2 +6^2[/tex] = 10
-2 +36 = 10
34 = 10 which is not true
if u = 1 ad v = 3
then;
[tex]u +v^2[/tex] = 10
[tex]1 +3^2[/tex] = 10
1 +9 = 10
10 = 10 which is not true.
Therefore, the point q lies on the given surface with u=1 and v =3
