The vertex A of triangle ABC is on the line r→=i^+j^+λk^ and the vertices B and C have respective position vectors i^ and j^ let ∆ be the area of the triangle and Δ∈[3/2,33/2] Then the range of values of λcorresponding to I is

Δ=12|(j^+λk^)×(i^+λk^)|=12|−k^+λi^+λj^| =122λ2+1⇒ 94≤142λ2+1≤334 or 4≤λ2≤16 or 2≤|λ|≤4

The vertex A of triangle ABC is on the line r→=i^+j^+λk^ and the vertices B and C have respective position vectors i^ and j^ let ∆ be the area of the triangle and Δ∈[3/2,33/2] Then the range of values of λcorresponding to I is

Δ=12|(j^+λk^)×(i^+λk^)|=12|−k^+λi^+λj^| =122λ2+1⇒ 94≤142λ2+1≤334 or 4≤λ2≤16 or 2≤|λ|≤4

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The vertex A of triangle ABC is on the line $\vec{r} = \hat{i} + \hat{j} + λ \hat{k}$ and the vertices B and C have respective position vectors $\hat{i} and \hat{j}$ let $∆$ be the area of the triangle and $Δ \in [3 / 2, \sqrt{33} / 2]$ Then the range of values of $λ$ corresponding to I is

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$[- 4, 4]$

$[- 8, - 4] \cup [4, 8]$

$[- 2, 2]$

$[- 4, - 2] \cup [2, 4]$

answer is D.

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Detailed Solution

$\begin{array}{l} \begin{aligned} Δ = \frac{1}{2} | (\hat{j} + λ \hat{k}) \times (\hat{i} + λ \hat{k}) | & = \frac{1}{2} | - \hat{k} + λ \hat{i} + λ \hat{j} | \\ = \frac{1}{2} \sqrt{2 λ^{2} + 1} \end{aligned} \\ \Rightarrow \frac{9}{4} \leq \frac{1}{4} (2 λ^{2} + 1) \leq \frac{33}{4} \\ or 4 \leq λ^{2} \leq 16 \\ or 2 \leq | λ | \leq 4 \end{array}$