Find the equation of a plane passing through the pint `P(6,\ 5,\ 9)` and parallel to the plane determined by the points `A(3,\ -1,\ 2)B(5,\ 2,\ 4)a n d\ C(-1,\ -1,\ 6)dot` Also find the distance of this plane from the point A.

Explanation

Let


`vec{A B}=5 hat{i}+2 hat{j}+4 hat{k}-3 hat{i}+hat{j}-2 hat{k}=2 hat{i}+3 hat{j}+2 hat{k} `

`vec{B C}=-hat{i}-hat{j}+6 hat{k}-5 hat{i}-2 hat{j} -6 hat{i}-3 hat{j}+2 hat{k}`


Normal is determined by the points `A, B` and `C` is `vec{m}=vec{A B} times vec{B C}`

`therefore vec m = |[hat i, hat j, hat k],[2,3,2],[-6,-3,2]|` `= 12 hat i – 16 hat j + 12 hat k`

`therefore` normal to the requires `pi = 3,-4,3`

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