Question -
Answer -
(a) 2x + 3y + 4z – 12 = 0
Let the coordinate ofthe foot of ⊥ P from theorigin to the given plane be P(x, y, z).
2x + 3y + 4z = 12 ….(1)
Direction ratio are(2, 3, 4)
√[(2)2 +(3)2 + (4)2] = √(4 + 9 + 16)
= √29
Now,
Divide both the sidesof equation (1) by √29, weget
2x/(√29) + 3y/(√29) + 4z/(√29) =12/√29
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
∴ The direction cosinesare 2/√29, 3/√29, 4/√29
Coordinate of the foot(ld, md, nd) =
= [(2/√29) (12/√29), (3/√29) (12/√29), (4/√29) (12/√29)]
= 24/29, 36/29, 48/29
(b) 3y + 4z – 6 = 0
Let the coordinate ofthe foot of ⊥ P from theorigin to the given plane be P(x, y, z).
0x + 3y + 4z = 6 ….(1)
Direction ratio are(0, 3, 4)
√[(0)2 +(3)2 + (4)2] = √(0 + 9 + 16)
= √25
= 5
Now,
Divide both the sidesof equation (1) by 5, we get
0x/(5) + 3y/(5) +4z/(5) = 6/5
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
∴ The direction cosinesare 0/5, 3/5, 4/5
Coordinate of the foot(ld, md, nd) =
= [(0/5) (6/5), (3/5)(6/5), (4/5) (6/5)]
= 0, 18/25, 24/25
(c) x + y + z = 1
Let the coordinate ofthe foot of ⊥ P from theorigin to the given plane be P(x, y, z).
x + y + z = 1 …. (1)
Direction ratio are(1, 1, 1)
√[(1)2 +(1)2 + (1)2] = √(1 + 1 + 1)
= √3
Now,
Divide both the sidesof equation (1) by √3, weget
1x/(√3) + 1y/(√3) + 1z/(√3) = 1/√3
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
∴ The direction cosinesare 1/√3, 1/√3, 1/√3
Coordinate of the foot(ld, md, nd) =
= [(1/√3) (1/√3), (1/√3) (1/√3), (1/√3) (1/√3)]
= 1/3, 1/3, 1/3
(d) 5y + 8 = 0
Let the coordinate ofthe foot of ⊥ P from theorigin to the given plane be P(x, y, z).
0x – 5y + 0z = 8 ….(1)
Direction ratio are(0, -5, 0)
√[(0)2 +(-5)2 + (0)2] = √(0 + 25 + 0)
= √25
= 5
Now,
Divide both the sidesof equation (1) by 5, we get
0x/(5) – 5y/(5) +0z/(5) = 8/5
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
∴ The direction cosinesare 0, -1, 0
Coordinate of the foot(ld, md, nd) =
= [(0/5) (8/5), (-5/5)(8/5), (0/5) (8/5)]
= 0, -8/5, 0