Let a line passing through the point ( - 1 , 2 , 3 ) intersect the lines L 1 : x - 1 3 = y - 2 2 = z + 1 - 2 at M ( α , β , γ ) and L 2 : x + 2 - 3 = y - 2 - 2 = z - 1 4 at N ( a , b , c ) . Then the value of ( α + β + γ ) 2 ( a + b + c ) 2

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Let a line passing through the point ( - 1 , 2 , 3 ) intersect the lines L 1 : x - 1 3 = y - 2 2 = z + 1 - 2 at M ( α , β , γ ) and L 2 : x + 2 - 3 = y - 2 - 2 = z - 1 4 at N ( a , b , c ) . Then the value of ( α + β + γ ) 2 ( a + b + c ) 2 equals ________________.

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Let, L 1 : x - 1 3 = y - 2 2 = z + 1 - 2 = λ ⇒ M ( 3 λ + 1 , 2 λ + 2 , - 2 λ - 1 ) ≡ M α , β , γ ⇒ α + β + γ = 3 λ + 2 Also, L 2 : x + 2 - 3 = y - 2 - 2 = z - 1 4 = μ ⇒ N ( - 3 μ - 2 , - 2 μ + 2 , 4 μ + 1 ) ≡ N a , b , c ⇒ a + b + c = - μ + 1 Now, from the above diagram, we can see M N lie on the same line, So, 3 λ + 2 - 3 μ - 1 = 2 λ - 2 μ = - 2 λ - 4 4 μ - 2 Now, solving 3 λ + 2 - 3 μ - 1 = 2 λ - 2 μ ⇒ 3 λ μ + 2 μ = 3 λ μ + λ ⇒ 2 μ = λ . . . . . . i Now, solving 2 λ - 2 μ = - 2 λ - 4 4 μ - 2 we get, ⇒ 2 λ μ - λ =

Let a line passing through the point ( - 1 , 2 , 3 ) intersect the lines L 1 : x - 1 3 = y - 2 2 = z + 1 - 2 at M ( α , β , γ ) and L 2 : x + 2 - 3 = y - 2 - 2 = z - 1 4 at N ( a , b , c ) . Then the value of ( α + β + γ ) 2 ( a + b + c ) 2 equals ________________.

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