Exams Joint Entrance Examination Mathematics Vector Algebra Question 18 Question 18 of 379 Medium MCQ
Let A B → ← = 2 i ^ + 4 j ^ − 5 k ^ \overset{\to \leftarrow }{AB}=2\hat{i}+4\hat{j}-5\hat{k} A B →← = 2 i ^ + 4 j ^ − 5 k ^ and A D → ← = i ^ + 2 j ^ + λ k ^ , λ ∈ R \overset{\to \leftarrow }{AD}=\hat{i}+2\hat{j}+\lambda \hat{k},\lambda \in R A D →← = i ^ + 2 j ^ + λ k ^ , λ ∈ R . Let the projection of the vector v ⃗ = i ^ + j ^ + k ^ \vec{v}=\hat{i}+\hat{j}+\hat{k} v = i ^ + j ^ + k ^ on the diagonal A C → ← \overset{\to \leftarrow }{AC} A C →← of the parallelogram ABCD be of length one unit. If α , β \alpha ,\beta α , β , where α > β \alpha >\beta α > β , be the roots of the equation λ 2 x 2 − 6 λ x + 5 = 0 {\lambda }^{2}{x}^{2}-6\lambda x+5=0 λ 2 x 2 − 6 λ x + 5 = 0 , then 2 α − β 2\alpha -\beta 2 α − β is equal to
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A B ⃗ = 2 i ⃗ + 4 j ⃗ − 5 k ⃗ A D ⃗ = i ⃗ + 2 j ⃗ + λ k ⃗ V ⃗ = i ⃗ + j ⃗ + k ⃗ V ⃗ ⋅ A C ⃗ ∣ A C ⃗ ∣ \begin{gathered} \vec{AB}=2\vec{i}+4\vec{j}-5\vec{k} \\ \vec{AD}=\vec{i}+2\vec{j}+\lambda \vec{k} \\ \vec{V}=\vec{i}+\vec{j}+\vec{k} \\ \vec{V}\cdot \vec{AC} \\ \frac{|\vec{AC}|}{} \end{gathered} A B = 2 i + 4 j − 5 k A D = i + 2 j + λ k V = i + j + k V ⋅ A C ∣ A C ∣
A C ⃗ = A B ⃗ + B C ⃗ 18 λ = 54 18 λ = 54 \begin{gathered} \vec{AC}=\vec{AB}+\vec{BC} \\ 18\lambda =54 \\ 18\lambda =54 \end{gathered} A C = A B + B C 18 λ = 54 18 λ = 54 Answer the question to unlock