find the points at which the line`(x-6)/3=(y+2)/-6=(z-2)/4` intersects ellipsoid`x^2/81+y^2/36+z^2/9=1`

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`(x-6)/3= (y+2)/-6`

`-6x+36 = 3y + 6` ` `

`x = -1/2y+5`

`(y+2)/-6 = (z-2)/4`

`4y+8 = -6z+12`

`z = -2/3y+2/3`

`((-1/2y+5)^2)/81 + y^2/36 + ((-2/3y+2/3)^2)/9 = 1`

Multiply by 324

`4(-1/2y+5)^2 + 9y^2 + 36(-2/3y+2/3)^2=324`

`4(1/4y^2-5y+25) + 9y^2 + 36(4/9y^2 -8/9y+4/9)=324`

`y^2 - 20y + 100 + 9y^2 + 16y^2 -32y + 16 = 324`

`26y^2-52y+116 = 324`

`26y^2 - 52y - 208 = 0`

`y^2-2y-8=0`

`(y - 4)(y + 2) = 0`

`y = 4, y = -2`

`(x-6)/3 = (4+2)/-6 = (z - 2)/4`

`(x - 6)/3 = -1` and `-1 = (z-2)/4`

`x = 3 and z = -2`

This makes 1 solution `(3, 4, -2)`

`(x-6)/3 = (-2+2)/-6 = (z - 2)/4`

`(x-6)/3 = 0 = (z-2)/4`

`x = 6` and `z = 2`

This makes 2nd solution `(6, -2, 2)`

The line intersects the ellipsoid at (3, 4, -2) and (6, -2, 2).

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