Vector Calculus - Equations for planes tangent to given equation

[ysolidusx]

Homework Statement

My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x² + 6x + y³ that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states to find "Equation-S".
Variables are x, y, and z

Homework Equations

To reiterate:
z = x² + 6x + y³
4x − 12y + z = 7
General form of the graph of a tangent plane
z = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b)

The Attempt at a Solution

I understand that in order to find a tangent plane to a particular point on the graph of some function one must compute the partial derivatives of the original function and then compute the remaining portion from the point that is given.

z_x = 2x - 6
z_y = 3y²

What is confusing me is the part about the plural. Is the plural part hanging me up and getting in the way, or do I use the normal vector to attempt to find a point on the original plane?
In addendum: I would like to apologize that my equations aren't in the proper TeX format. I'm only just getting the hang of writing TeX scripts.

 

[tiny-tim]

welcome to pf!

hi ysolidusx! welcome to pf!

"Find equations for the planes tangent to z = x² + 6x + y³ that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect.

eg, if it was a sphere, there would be two tangent planes (on opposite sides of the sphere) parallel to any given plane …

you have to find all of them! ​

 

[ysolidusx]

So finding all of the planes would be akin to finding all of the lines through the origin by creating the general form of the equation:
y = Ax + B
Where A and B could be any constants and thus determine any line.

Thus to find the tangent planes, you would find a general form of the equation and then find those subsets that satisfy the constraint of being parallel to the given plane.

Thank you very much for the quick return on the message there. That explanation is quite simple but very helpful. I do greatly appreciate your assistance.