May 9th, 2017, 01:33 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Value of λ
Hello All Can anyone help me to get the process of solving this issue: Find the value of $\lambda$ such that the plane $2xy+\lambda z=0$ is a tangent plane to the sphere $x^2+y^2+z^22x2y2z+2=0$ is (A) 4 $\ $ (B) 1 $\ $ (c) 2 $\ $ (D) 2 Last edited by skipjack; May 9th, 2017 at 01:37 AM. 
May 9th, 2017, 02:22 AM  #2 
Senior Member Joined: Feb 2010 Posts: 679 Thanks: 127 
C

May 9th, 2017, 02:30 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,054 Thanks: 1618 
Consider the point (1/3, 4/3, 1/3) on the sphere.

May 9th, 2017, 02:54 AM  #4 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
May 9th, 2017, 09:32 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,191 Thanks: 871 
First, Complete the squares in the equation for the sphere to write the sphere in "standard form" so you can determine the center and radius of the sphere. Second, note that the normal vector to the plane Ax+ By+ Cz= D is <A, B, C>. The line through the center of the sphere in the direction of that normal vector intersects the plane at the point where the plane is tangent to the sphere.

May 9th, 2017, 03:10 PM  #6 
Member Joined: Feb 2015 From: Southwest Posts: 96 Thanks: 24 
Could you use Lagrange multipliers to solve this? I feel like no because that would give you 5 unknowns and only 4 equations, right? I've been trying to solve this, and I keep hitting road blocks. I'm not sure what method should be employed.

May 9th, 2017, 09:32 PM  #7  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
Please help!!!  
May 9th, 2017, 11:00 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,054 Thanks: 1618 
A normal vector to the plane $2x  y + \lambda z = 0$ is (2, 1, $\lambda$) (or <2, 1, $\lambda$> if you prefer that notation). If you guess that $\lambda$ = 2, what is the length of that vector? You don't have to make that guess, but a lucky guess can simplify what you still need to do.

May 10th, 2017, 01:19 AM  #9  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
$Ax+By+Cz+D=0$ $2(1)1+$ $\lambda$ $(1)+1=0$ $21+$ $\lambda$ $+1=0$ $2+$ $\lambda$ $=0$ $\lambda$ $= 2$ then the answer would be option $ D$ : $2$ Please help me if I'm wrong in the process.  
May 10th, 2017, 03:54 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 19,054 Thanks: 1618 
A line through the point (1, 1, 1) parallel to that normal vector intersects the plane at the point where the plane is tangent to the sphere. How far will that intersection point (assuming it exists) be from the point (1, 1, 1)? Hence what is the intersection point?
