Does the line with parametric equation `x=2t+1`, `y=3t-2`, `z=2t-2`, teR intersect the sphere `x^2+y^2+z^2=9`? If so, determine the points of intersection.

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You need to substitute `2t + 1 , 3t - 2 , 2t -2`  for x,y,z in equation of the sphere such that:

`(2t+1)^2 + (3t - 2)^2 + (2t - 2)^2 = 9`

You need to expand binomials such that:

`4t^2 + 4t + 1 + 9t^2 - 12t...

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You need to substitute `2t + 1 , 3t - 2 , 2t -2`  for x,y,z in equation of the sphere such that:

`(2t+1)^2 + (3t - 2)^2 + (2t - 2)^2 = 9`

You need to expand binomials such that:

`4t^2 + 4t + 1 + 9t^2 - 12t + 4 + 4t^2 - 8t + 4 = 9`

You need collect like terms and you need to sovle for t the quadratic equation `17t^2 - 16t = 0` .

You need to factor out t such that:

`t(17t - 16) = 0`

`t = 0`

`17t-16-0 =gt t = 16/17`

Since, the equation has two solution the line will intersect the sphere at `t=0`  and `t=16/17` .

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