Let P denotes the plane consisting of all points that are equidistant from the points `A(-4,2,1)` and `B(2,-4,3)` and Q be the plane, `x-y+cz=1` where `c in R`.
If the angle between the planes P and Q is `45^(@)` then the product of all possible values of c is

Options

A. `-17`
B. `-2`
C. 17
D. `24//27`

Answer

B

Explanation

`vecn_(1) = 3hati-3hatj+hatk` and `vecn_(2) = hati-hatj+chatk`
`cos 45^(@) = |(vecn_(1).vecn_(2))/(|vecn_(1)||vecn_(2)|)|=|((3+3+c)/(sqrt(19)sqrt(2+c^(2))))|`
`therefore 2(6+c)^(2) = 19(2+c^(2))`
`rArr 2(36+c^(2)+12c)=38+19c^(2)`
`rArr 17c^(2)-24c-34=0`