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October 6th, 2017, 02:44 AM  #1 
Newbie Joined: Oct 2017 From: Sweden Posts: 3 Thanks: 0  A question regarding how to define a plane
Today I took a math test and one of the questions was: "Prove that 4x + 3y = 7 is a tangent plane to x^2 + 2xy + y = 4 and state the tangent point." This question confused me, since the question didn't state that x or y were functions of any variables. Therefore I'm thinking that both of the equations are in fact not planes (3 dimensions), but graphs (2 dimensions). I'm also thinking that there is no way to get a (x,y,z) point, since the functions only consist of 2 variables. Am I thinking completely wrong or is the question incorrectly stated? Also, my first language is not English so forgive me if I am using the wrong terms. Thanks! 
October 6th, 2017, 04:55 AM  #2  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,115 Thanks: 708 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
The problem is basically a 2D problem since you don't have to worry about the zcoordinate, but there's nothing wrong with using those equations to describes planes or surfaces in 3D space. You can get 3D coordinates without any problem, but in a problem with no zdependence, the zcomponent of any coordinate you find will probably be 0. Last edited by Benit13; October 6th, 2017 at 04:57 AM.  
October 6th, 2017, 05:08 AM  #3  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,196 Thanks: 872  Quote:
Quote:
You can solve this as "Show that 4x+ 3y= 7 is the tangent line to x^2+ 2xy+ y= 4" in the xyplane. That (x, y) point, with z any number, would be the line of tangency of the plane and surface. If it were me, I would double check to make certain there was not supposed to be a "z" in at least one of those equations.  
October 6th, 2017, 05:29 AM  #4 
Newbie Joined: Oct 2017 From: Sweden Posts: 3 Thanks: 0 
Thank you for your replies. There was no "z" in the equation, so I was thrown off by that. I guess I just have a tricky teacher that hands out this problem on a test At least now I know that I should have treated it as equations in the xyplane. The extra tricky thing is that we learned a formula for getting a tangent plane and it contained z and f derived with respect to x and y. 
October 6th, 2017, 07:18 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,059 Thanks: 1619 
Note that x = y = 1 satisfies both equations.

October 6th, 2017, 08:47 AM  #6 
Newbie Joined: Oct 2017 From: Sweden Posts: 3 Thanks: 0  

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