Question

If tan, cot & are roots of x² + 2ax + b = 0, then least value of |a|
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Answer 1

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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