     1. Chapter 11 Class 12 Three Dimensional Geometry
2. Serial order wise
3. Miscellaneous

Transcript

Misc 23 (Method 1) The planes: 2x y + 4z = 5 and 5x 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through 0,0, 5 4 Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos = + + + + + + Given the two planes are So, cos = 2 10 + 1 5 + (4 20) 2 2 + ( 1) 2 + 4 2 10 2 + ( 5) 2 + 20 2 = 20 + 5 + 80 4 + 1 + 16 100 + 25 + 400 = 105 21 525 = 105 21 25 21 = 105 21 5 21 = 105 21 5 = 1 So, cos = 1 = 0 Since angle between the planes is 0 , Therefore, the planes are parallel. So, option (B) is correct Misc 23 (Method 2) The planes: 2x y + 4z = 5 and 5x 2.5y + 10z = 6 are (A) Perpendicular (B) Parallel (C) intersect y-axis (D) passes through 0,0, 5 4 Given, two planes are Two lines are parallel if their direction ratios are proportional. 1 2 = 2 10 = 1 5 , 1 2 = 1 5 = 1 5 , 1 2 = 4 20 = 1 5 Since, = = = , Therefore, the normal vectors of the two planes are parallel. So, the two planes are parallel. So, option (B) is correct

Miscellaneous 