1.

Given that tan α and tan β  are the roots of the equation 2x2 + bx + c = 0. What is the cot (α + β) equal to 1. \(\rm \frac{-b}{2-c}\)2. \(\rm \frac{c-2}{b}\)3. 1/(b - c)4. 2/(b - c)

Answer» Correct Answer - Option 2 : \(\rm \frac{c-2}{b}\)

Concept:

Consider a quadratic equation: ax2 + bx + c = 0.

Let, α and β are the roots.

  • Sum of roots = α + β = -b/a
  • Product of the roots = α × β = c/a
  • tan(A + B) = \(\rm \frac{tanA+tanB}{1-tanAtanB}\)

 

Calculation:

Here, tanα and tanβ  are the roots of the equation 2x2 + bx + c = 0

Sum of roots =  tan α + tan β = -b/2

Product of roots = c/2

tan (α + β) = \(\rm \frac{tan\alpha +tan\beta }{1-tan\alpha tan\beta}\)

= \(\rm \frac{\frac{-b}{2}}{1-\frac c 2}\)

= \(\rm \frac{-b}{2-c}\)

= \(\rm \frac{b}{c-2}\)

cot (α + β) = \(\rm \frac{c-2}{b}\)

Hence, option (2) is correct.


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