Find the distance

Geometry Level 5

The shortest distance from ( 1 , 1 , 1 ) 1,1,1) to the line of intersection of the pair of planes x y + y z + z x + y 2 = 0 xy+yz+zx+y^2=0 is


The answer is 1.633.

Tanishq Varshney by Tanishq Varshney , 23, India

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1 solution

Julian Poon
Mar 31, 2015

First, let's manipulate the equation to make it more manageable.

x y + y z + z x + y 2 = 0 xy+yz+zx+y^2=0

( x + y ) ( z + y ) = 0 (x+y)(z+y)=0

So, 1 1 plane would be when x = − y x=-y and the other plane would be when z = − y z=-y

The line of intersection would be when both conditions are satisfied, as in x = z = − y x=z=-y

hi hi

This simplifies the whole question:

hello again hello again

L L can then be easily calculated to be 2 2 3 \frac{2\sqrt{2}}{\sqrt{3}}

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