Show that two surfaces intersect at the given point

Question

The given surfaces and point are $$f(x,y)=xy$$ $$g(x,y)=\frac{3}{4}x^2-2y$$ $$(2,-3,-6)$$ First I made each surface have three variables by subtracting $z$ from both sides. Then I found the gradient of the surfaces $$f(x,y,z)=\left<y,x,-1\right>$$ $$g(x,y,z)=\left<\frac{3}{2}x, -2y ,-1 \right>$$ Then finding the gradient at the given point $$f(x,y,z)=\left< 2,-3,-1\right>$$ $$g(x,y,z)=\left<3,6 ,-1 \right>$$ Then I found the cross product of the two and got $$9i -j+21k$$ But I'm not sure how this shows that the two surfaces intersect.

Answer 1

Hint The surfaces do not intersect at the point in OP because that point does not belong to the second surface. They do, however, have the whole curve $y=3/4x^2/(x+2),z=xy$ in common. In particular , they have the point $(4,2, 8)$ in common. Proving that the surfaces intersect (and not touch) at that point can be done along the line you suggested.