6 )
For A(7, – 3, 1) and B(4, 9, 8), the point that divides AB from B in the ratio 2:5 is ____
[(34 / 7), (39 / 7), (42 / 7)] ✔
[(34 / 7), (39 / 7), {(– 42) / 7}]
[{(– 34) / 7}, (39 / 7), {(– 42) / 7}]
[{(– 34) / 7}, {(– 39) / 7}, {(– 42) / 7}]
View Solution
Answer :
A(7, – 3, 1) and B(4, 9, 8)
Let P(x, y, z) be the dividing point.
∴ (x, y, z) = [{[2(7) + 5(4)] / (2 + 5)}, {[2(– 3) + 5(9)] / (2 + 5)}, {[2(1) + 5(8)] / (2 + 5)}]
= [(34 / 7), (39 / 7), (42 / 7)]
7 )
For A(1, 5, 6), B(3, 1, 2) and C(4, – 1, 0), B divides AC from A in ____ ratio
– 2 : 3
2 : 3
2 : 1 ✔
– 2 : 1
View Solution
Answer :
Let B divides AC in the ratio λ : 1
A(1, 5, 6), B(3, 1, 2) and C(4, – 1, 0)
∴ 3 = [(4λ + 1) / (λ + 1)], 1 = [(– λ + 5) / (λ + 1)], 2 = [(0 + 6) / (λ + 1)]
∴ 3λ + 3 = 4λ + 1, λ + 1 = – λ + 5 and 2λ + 2 = 6
∴ λ = 2, λ = 2, λ = 2
ratio: 2 : 1
8 )
A(0, – 1, 4), B(1, 2, 3), C(5, 4, – 1), then the foot of perpendicular from A on BC is ____
(– 3, 3, 1)
(3, – 3, 1)
(3, 3, 1) ✔
(3, 3, – 1)
View Solution
Answer :
A(0, – 1, 4), B(1, 2, 3) and C(4, – 1, 0)
Let D divides BC from B in the ratio λ : 1 then
D = [{(5λ + 1) / (λ + 1)}, {(4λ + 2) / (λ + 1)}, {(– λ + 3) / (λ + 1)}]
also BC = (4, 2, – 4)
AD = [{(5λ + 1) / (λ + 1)}, {(5λ + 3) / (λ + 1)}, {(3λ + 7) / (λ + 1)}]
Now BC ⊥ AD
⇒ BC ∙ AD = 0
∴ 4[(5λ + 1) / (λ + 1)] + 2[(5λ + 3) / (λ + 1)] + (– 4) [(3λ + 7) / (λ + 1)] = 0
∴ 20λ + 4 + 10λ + 6 – 12λ – 28 = 0
∴ 18λ = 18
∴ λ = 1
∴ D = (3, 3, 1)
9 )
If A(a, 1, 3), B(– 1, b, 2), C(1, 0, c) are the vertices of ΔABC whose centroid is (2, 3, 5), then values of a, b, c are respectively ____
10, 8, 6
6, 10, 8
8, 6, 10
6, 8, 10 ✔
View Solution
Answer :
given: A(a, 1, 3), B(– 1, b, 2) and C(1, 0, c) and centroid is given by
(2, 3, 5)
∴ (2, 3, 5) = [{(a – 1 + 1) / 3}, [(1 + b + 0) / 3}, {(3 + 2 + c) / 3}]
∴ (a/3) = 2, [(b + 1) / 3] = 3, [(5 + c) / 3] = 5
∴ a = 6, b = 8, c = 10
10 )
If A(6, 4, 6), B(12, 4, 0), C(4, 2, – 2) are the vertices of triangle, then it's in centre is ____
[(22 / 3), (10 / 3), (4/3)] ✔
[{(– 22) / 3} (10 / 3), (4/3)]
[(22 / 3), {(– 10) / 3}, (4/3)]
[(22 / 3), (10 / 3), {(– 4) / 3}]
View Solution
Answer :
A(6, 4, 6), B(12, 4, 0) and C(4, 2, – 2)
Now AB = √(36 + 0 + 36) = √72 = c
BC = √(64 + 4 + 4) = √72 = a
AC = √(4 + 4 + 64) = √72 = b
∴ ΔABC is equilateral triangle
⇒ incentre and centroid are equal
∴ centroid = [{(6 + 12 + 4) / 3}, {(4 + 4 + 2) / 3}, {(6 + 0 – 2) / 3}]
= [(22 / 3), (10 / 3), (4/3)]