1. ## Tangential surfaces

Hey, guys. So I'm about to have a test in my calc class and I'm working through the review that my professor handed out. I really need to know whether I'm doing some of these problems right before test day arrives, so if you guys don't mind, would you check my work and answers for some questions?

Here's one: "Two surfaces are said to be tangential at a point P if (1), the point P is on both surfaces and (2), the two surfaces have the same tangent plane at P. Show that the surfaces $(2x^2)+(2y^2)-(z^2)=100$ and $z=(1/10)(x^2+y^2)$ are tangential at the point (8,6,10)."

Here's my work:

a) For $(2x^2)+(2y^2)-(z^2)=100$, I found fx, fy, and fz, which are 4x, 4x, and -2z.

Then I plugged the points (8,6,10) into that, so f(8,6,10) = (32,24,-20).

So the equation of this plane becomes $32(x-8)+24(y-6)-20(z-10)=0$, which condenses to $8x+6y-5z=50$.

b) For $z=(1/10)(x^2+y^2)$, I went through the same process.

fx, fy, and fz = (1/5)x, (1/5)y, -1

f(8,6,10)= (8/5, 6/5, -1)

Equation of the plane: $(8/5)(x-8)+(6/5)(y-6)-(z-10)=0$,
which also condenses to $8x+6y-5z=50$.

Therefore, these two surfaces are tangential.

Is this right?

2. That does work. But there is a much shorter way.
Show that the point is no both surfaces.
Show that the two gradients at the point are parallel.

3. Hm... Okay, I see. Thanks!