If →b is a vector whose initial point divides the join of 5^i and 5^j in the ratio k:1 and whose terminal point is the origin and |→b|≤√37, then k lies in the interval
A
[−6,−16]
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B
(−∞,−6)∪[−16,∞)
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C
[0,6]
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D
None of these
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Solution
The correct option is B(−∞,−6)∪[−16,∞) The point that divides 5^i and 5^j in the ratio of k:1 is =(5^j)k+(5^i)1k+1 ∴→b=−(5^i+5k^jk+1)
Also, |→b|≤√37 ⇒1k+1√25+25k2≤√37 ⇒5√1+k2≤√37(k+1) Squaring both sides, we get 25(1+k2)≤37(k2+2k+1) or, 6k2+37k+6≥0 or, (6k+1)(k+6)≥0 or, k∈(−∞,−6]∪[−16,∞)