# If tanα.tanβ = a and α + β = π/6 then tanα and tanβ are the roots of the quadratic equation: a. sqrt(3)x^2 - (1 - a)x +sqrt(3)a = 0 b. sqrt(3)x^2 - (1 + a)x + sqrt(3)a = 0 c. sqrt(3)x^2 + (1...

If tanα.tanβ = a and α + β = π/6 then tanα and tanβ are the roots of the quadratic equation:

a. sqrt(3)x^2 - (1 - a)x +sqrt(3)a = 0

b. sqrt(3)x^2 - (1 + a)x + sqrt(3)a = 0

c. sqrt(3)x^2 + (1 - a)x - sqrt(3)a = 0

d. sqrt(3)x^2 + (1 - a)x + sqrt(3)a = 0

### 2 Answers | Add Yours

a) First divide the equations by sqrt(3)

x^2 - (1-a)x /sqrt(3) - a = 0

so: tg`alpha` + tg `beta` = 1- tg `beta`tg`alpha` /sqrt(3)

thus (tg`alpha` +tg`beta` )/( 1 -tg`alpha` tg`beta` ) = sqrt(3)/3

that is: tg( `alpha`+ `beta` )= sqrt(3)/3

and verifies for + `` = `pi` /6

on the other side a = tg`alpha` tg`beta`

so are the solutions.

b) x^2 - (1 + a) x /sqrt(3)+ a = 0

At the same: tg `alpha` +tg `beta` = (1 +tg`alpha` tg`beta` )/sqrt(3)

so [that tg `alpha` -(-tg`beta` )]/ [1 -tg`alpha` (-tg`beta` )] =sqrt(3)/3

tg(`alpha` + `pi` -`beta` )=sqrt(3) /3

tg(5`pi` /6) = sqrt(3)/3 not verified

c) tg`alpha` +tg`beta` =-(1 - tg`alpha` tg`beta` )/sqrt( 3)

then:

-(tg `alpha` + tg `beta` ) =-( 1 - tg`alpha` tg`beta` )/sqrt(3)

tg(`alpha` +`beta` )= sqrt(3)/3 so is veriifed

d) - (tg`alpha` +tg`beta` )= 1 - tg`alpha` tg`beta` /sqrt(3)

so (as the point "c", :

- tg(`alpha` +`beta` )= sqrt(3)/3

tg[`pi` -(`alpha` +`beta` )]=tg(5`pi` /6)= sqrt(3)/3

not verified

`tan(alpha+beta)=(tan (alpha)+tan(beta))/(1-tan(alpha)tan(beta))`

`tan(pi/6)=(x+y)/(1-a)`

where x=`tan(alpha) ,y=tan(beta)`

`x+y=(1-a)/sqrt(3)`

`xy=a`

Ans. a.

sum of the roots=(1-a)/sqrt(3)

product of the roots= a