Let and , . Let the projection of the vector on the diagonal of the parallelogram ABCD be of length one unit. If , where , be the roots of the equation then is equal to [2026]

Question

Let $\vec{A B} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $\vec{A D} = \hat{i} + 2 \hat{j} + λ \hat{k}$ , $λ \in ℝ$ . Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\vec{A C}$ of the parallelogram ABCD be of length one unit. If $α, β$ , where $α > β$ , be the roots of the equation $λ^{2} x^{2} - 6 λ x + 5 = 0,$ then $2 α - β$ is equal to [2026]

Smart Achievers · Accepted Answer

(3)

$\vec{A C} = 3 \hat{i} + 6 \hat{j} + (λ - 5) \hat{k}$

$\vec{v} \cdot \vec{A C} = 1$ $\Rightarrow 3 + 6 + λ - 5 = \sqrt{9 + 36 + {(λ - 5)}^{2}}$

$\Rightarrow λ^{2} + 8 λ + 16 = λ^{2} - 10 λ + 70$

$\Rightarrow λ = \frac{54}{18} = 3$

$∴$ $Quadratic: 9 x^{2} - 18 x + 5 = 0$ $\Rightarrow x = \frac{1}{3}, \frac{5}{3}$

$∴$ $2 α - β = \frac{10 - 1}{3} = 3$

Solution

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Solution

Q. Let AB→=2i^+4j^-5k^ and AD→=i^+2j^+λk^, λ∈ℝ. Let the projection of the vector v→=i^+j^+k^ on the diagonal AC→ of the parallelogram ABCD be of length one unit. If α,β, where α>β, be the roots of the equation λ2x2-6λx+5=0, then 2α-β is equal to [2026]

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Ans. (3) AC→=3i^+6j^+(λ-5)k^ v→·AC→=1⇒3+6+λ-5=9+36+(λ-5)2 ⇒λ2+8λ+16=λ2-10λ+70 ⇒λ=5418=3 ∴Quadratic: 9x2-18x+5=0⇒x=13, 53 ∴2α-β=10-13=3

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