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. for no value of c
B. if c=3
C. if c`=1//3`
D. if c=1

Answer

C

Explanation

Plane P will contain the point `M(-1,-1,2)` and dr's of its normal will be proportional to `(3,-3,1)`.
Thus, equation of the plane P is
`3(x+1)-3(y+1)+1(z-)=0`
`therefore P:3x-3y+z-2=0`………..(1)
`therefore P:3x-3y+z-2=0`
Also, `Q:x-y+cz-1=0` ......(2)
If P is parallel to Q then
`3/1=1/c rArr c=1/3`.