If tan, cot & are roots of x² + 2ax + b = 0, then least value of |a|
Answers
Given : tan, cot & are roots of x² + 2ax + b = 0
To find : least value of |a| for real solutions
Solution:
tan α * Cot α = Product of roots = b
tan α * Cot α = 1
=> b = 1
=> x² + 2ax + 1 = 0
(2a)² - 4a ≥ 0 for real solution
=> 4a² - 4a ≥ 0
=> a(a - 1) ≥ 0
=> a ≥ 1 or a ≤ 0
tan α + Cot α = -2a
=> Sinα/Cosα + Cosα/Sinα = -2a
=> Sin²α + Cos²α = -2aSinαCosα
=> 1 = - aSin2α
=> a = - 1/Sin2α
Sin2α lies betwenn [ -1 , 1 ]
=> a ≤ - 1 or a ≥ 1
=> | a | ≥ 1
least value of | a | = 1
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Answer:
1
Step-by-step explanation:
the least value of |a|=1