The plane denoted by `P_(1) : 4x+7y+4z+81=0` is rotated through a right angle about its line of intersection with the plane `P_(2) : 5x+3y+ 10 z = 25`. If the plane in its new position is denoted by `P`, and the distance of this plane from the origin is d, then find the value of `[k//2]` (where `[*]` represents greatest integer less than or equal to k).

The plane denoted by `P_(1) : 4x+7y+4z+81=0` is rotated through a righ

1.	The plane denoted by `P_(1) : 4x+7y+4z+81=0` is rotated through a right angle about its line of intersection with the plane `P_(2) : 5x+3y+ 10 z = 25`. If the plane in its new position is denoted by `P`, and the distance of this plane from the origin is d, then find the value of `[k//2]` (where `[*]` represents greatest integer less than or equal to k).
Answer» Correct Answer - `7` `" "4x+7y+ 4z+ 81=0" "` (i) `" "5x+3y+10z=25" "`(ii) Equation of plane passing through their line of intersection is `(4x+7y+4z+81)+ lamda(5x+3y+10z-25)=0` or `" "(4+5lamda)x+ (7+3lamda)y+ (4+10lamda)z+ 81- 25lamda=0" "` (iii) Plane (iii) `bot` to (i), so `" "4(4+5lamda)+7(7+3lamda)+ 4(4+10lamda)=0` `therefore" "lamda=-1` From (iii), equation of plane is `" "-x+4y-6z+106=0" "`(iv) Distance of (iv) from (0, 0, 0)`=(106)/(sqrt(1+16+ 36))=(106)/(sqrt(53))`