Let z be a complex number satisfying |2z+10+10i|≤5√3−5. If arg denotes the principal argument lying in (−π,π], then the least value of arg(z) is
A
−5π6
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B
3π4
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C
−3π4
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D
5π6
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Solution
The correct option is A−5π6 |2z+10+10i|≤5√3−5 ⇒|z+(5+5i)|≤5√3−52 This represents interior of a circle whose centre is (−5,−5) and radius is 52(√3−1).
Here, AB=radius=52(√3−1) OA=√52+52=5√2 sinθ=52(√3−1)5√2=√3−12√2 ∴θ=15∘=π12 arg(z) is minimum when z is at B. ∠XOA=−π+tan−1∣∣∣−5−5∣∣∣=−3π4 ∠XOB=∠XOA+(−θ) =−3π4−π12=−5π6