May 19th, 2017, 01:30 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  Planes
The locus of a point $P$ which is at the same distance from two planes $x+y+z=1$ , $z=0$ is a) an unbounded set. b) a sphere. c) a pair of parallel planes. d) a pair of intersecting planes. Im guessing this could be intersecting of planes! Someone correct me If I'm wrong 
May 20th, 2017, 02:18 AM  #2 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
I think you are correct 
May 20th, 2017, 04:05 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,729 Thanks: 705 
The two given planes intersect along a straight line. Take any point along that straight line and construct the plane containing that point perpendicular to the two given planes. The two given planes intersect in this third plane in two straight lines intersecting at that point. That makes four angles, two pair of "vertical angles". Bisecting each pair gives the two planes that are "at the same distance from the two planes".

May 20th, 2017, 05:27 AM  #4 
Member Joined: Jan 2016 From: Athens, OH Posts: 51 Thanks: 27 
Analytically, this is easy. Let P=(x,y,z) be equidistant from the 2 planes. Then $${x+y+z1\over \sqrt3}=z$$ So $$(x+y+z1)^2=3z^2$$ Thus $$(x+y+z1\sqrt3z)(x+y+z1+\sqrt3z)=0$$ Hence $$P\text{ is on } x+y+(1\sqrt3)z1=0\text{ or P is on } x+y+(1+\sqrt3)z1=0$$ These two planes are not parallel since they have nonparallel normals; so they intersect. Conversely, if P is on either plane, the above steps are reversible and so P is equidistant from the original two planes. 

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