Consider three planes
P 1 ≡ 2x + y + z = 1
P 2 ≡ x – y + z = 2
P 3 ≡ α x + y + 3z = 5
The three planes intersects each other at point P on XOY plane and at point Q on YOZ plane. O is the origin.
Column –I | Column –II |
(i) The value of α is | [A] 1 |
(ii) The length of projection of PQ on x- axis is | [B] 2 |
(iii) If the coordinates of Point R situated at Minimum distance from point 'O' on the line PQ are (a, b, c), then the value of 7a + 14b +14c is | [C] 4 |
(iv) If the area of ΔPOQ is .Then the value of a – b is where a & b are co prime numbers | [D] 3 |
Text Solution
Verified by Experts(i) [C]; (ii) [A]; (iii) [B]; (iv) [D]
Ans.
(i) [C]
(ii) [A]
(iii) [B]
(iv) [D]
Sol.
(i) 
= 0 ⇒ α = 4
(ii) Put z = 0 in P 1 & P 2
2x + y = 1 and x – y = 2
∴ P ≡ (1, –1, 0)
Put x = 0 in P 1 & P 2
y + z = 1 and – y + z = 2
∴ Q ≡ [0, 1/2, 3/2]
= – 
Projection of PQ on x axis = 
= 1
(iii) 
Equation of line PQ is
= 
= 
= l
let R ≡ (– λ + 1, λ /2 – 1, 
)
. 
= 0
⇒ –1 ( λ + 1) + 
+ 
= 0
⇒ λ = 3/7
∴ R ≡ 
⇒ 7a + 14b + 14c = 2
(iv) Area = 
= 
∴ a – b = 3
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