Determine whether the points P and Q lie on the given surfac

Solution

For P we have u+v=4 u^2 - v = 2 u + v^2 = 6 Adding the first two we get u^2 + u - 6 =0 from where we get u= - 3 or u=2. As v = 4 - u, we have therefore v = 7 or v=2. If u= -3 and v=7 we have u + v^2 = 46 which is different from 6 but if u=v=2 we have u+v^2=6 and so the third equation is satisfied which means P lies on the surface with u=v=2 For Q the equations are u+v=5 u^2 - v = 1 u + v^2 = -11 Adding the first two we get u^2 + u - 6 =0 from where we get u= - 3 or u=2 and, as v = 5 -u, v=8 or v=3. None of these combinations satisfy u + v^2 =6 so Q can not lie on the surface