- #1

Easy_as_Pi

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## Homework Statement

Find the equations for two spheres that are tangent to the plane x+y-z=3 and x+y+z=9 and the line x=t , y=2t, z=3t passes through its center.

Preface: This problem was on a test I took yesterday. My professor handed it back today. The relevant equations and work are what he put on my test while grading it. So, I'm more trying to see why the method he chose to use is correct. This problem really threw me for a loop.

## Homework Equations

He has the distance from a point to a plane equation: d= |Ax1+By1+Cz1-D|/sqrt(A^2+B^2+C^2). I'm not sure why this equation was selected- I think it has to do with the fact that we know the center of both spheres, so it would be our point, and the distance from it to the plane would be the radius for each sphere?

## The Attempt at a Solution

His work is as follows:

x=t, y=2t, z=3t

(t,2t,3t) = center

t+2t-3t-3/sqrt3 = 3/sqrt3...not sure where the 3/sqrt3 on the right side came from. I know he just plugged in the t's to x+y-z=3, and then used the distance equation.

next I have: |6t-9|=3, solving for t we get t=2,1

plugging those into our center equation, we get (1,2,3) and (2,4,6)

Then, he just used the equation of a sphere, but had it equal to 3 for some reason.

(x-1)^2+(y-2)^2+(z-3)^2=3

(x-2)^2+(y-4)^2+(z-6)^2=3

Basically, if someone could guide me through this work a little bit, I would greatly appreciate it. The gears are turning, and the problem is starting to click, but I need a little push in the right direction.

Thanks!