My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Thanks Tree4Thanks
  • 1 Post By Lalitha183
  • 1 Post By agentredlum
  • 1 Post By Country Boy
  • 1 Post By johng40
Reply
 
LinkBack Thread Tools Display Modes
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
Thanks from agentredlum
Lalitha183 is offline  
 
May 20th, 2017, 02:18 AM   #2
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,372
Thanks: 233

I think you are correct

Thanks from Country Boy
agentredlum is offline  
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".
Thanks from Lalitha183
Country Boy is offline  
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
$${|x+y+z-1|\over \sqrt3}=|z|$$
So
$$(x+y+z-1)^2=3z^2$$
Thus
$$(x+y+z-1-\sqrt3z)(x+y+z-1+\sqrt3z)=0$$
Hence
$$P\text{ is on } x+y+(1-\sqrt3)z-1=0\text{ or P is on } x+y+(1+\sqrt3)z-1=0$$
These two planes are not parallel since they have non-parallel 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.
Thanks from Lalitha183
johng40 is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
planes



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Planes Swami Mathsananda Math 1 September 16th, 2015 12:53 PM
Planes and Spheres lilstef Geometry 10 June 23rd, 2014 10:00 AM
Planes and lines in 3D Cyberdollar Geometry 6 June 13th, 2014 11:04 AM
Planes gaussrelatz Algebra 3 October 7th, 2013 09:26 AM
Intersection of two planes in R3 majami Linear Algebra 1 October 2nd, 2010 10:18 AM





Copyright © 2017 My Math Forum. All rights reserved.