If b‾ is the vector whose initial point divides the joining 5i‾ and 5j‾ in the ratio λ:1 and terminal point is at origin. If |b→|≤37 then λ∈

b‾=5i^+5λj^λ+1 and b≤37 ⇒5i^+5λj^λ+1≤37 ⇒251+λ2λ+12≤37⇒25+25λ2≤371+λ2+2λ ⇒12λ2+74λ+12≥0 ⇒6λ2+37λ+6≥0 ⇒6λ+1|λ+6≥0⇒λ∈(-∞,-6]∪-16,∞

If b‾ is the vector whose initial point divides the joining 5i‾ and 5j‾ in the ratio λ:1 and terminal point is at origin. If |b→|≤37 then λ∈

b‾=5i^+5λj^λ+1 and b≤37 ⇒5i^+5λj^λ+1≤37 ⇒251+λ2λ+12≤37⇒25+25λ2≤371+λ2+2λ ⇒12λ2+74λ+12≥0 ⇒6λ2+37λ+6≥0 ⇒6λ+1|λ+6≥0⇒λ∈(-∞,-6]∪-16,∞

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If $\overline{b}$ is the vector whose initial point divides the joining $5 \overline{i}$ and $5 \overline{j}$ in the ratio $λ : 1$ and terminal point is at origin. If $| \vec{b} | \leq \sqrt{37}$ then $λ \in$

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$(- \infty, 0) \cup (\frac{1}{2}, \infty)$

$⌈- 6, - \frac{1}{6}⌉$

$(- \infty, - 3) \cup [- \frac{1}{4}, \infty)$

$(- \infty, - 6] \cup [- \frac{1}{6}, \infty)$

answer is A.

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Detailed Solution

$\overline{b} = \frac{5 \hat{i} + 5 λ \hat{j}}{λ + 1}$ and $|b| \leq \sqrt{37} \Rightarrow |\frac{5 \hat{i} + 5 λ \hat{j}}{λ + 1}| \leq \sqrt{37} \Rightarrow \frac{25 (1 + λ^{2})}{{(λ + 1)}^{2}} \leq 37 \Rightarrow 25 + 25 λ^{2} \leq 37 (1 + λ^{2} + 2 λ) \Rightarrow 12 λ^{2} + 74 λ + 12 \geq 0 \Rightarrow 6 λ^{2} + 37 λ + 6 \geq 0 \Rightarrow (6 λ + 1) (| λ + 6) \geq 0 \Rightarrow λ \in (- \infty, - 6] \cup [- \frac{1}{6}, \infty)$