If `a` is real constant `A ,B and C` are variable angles and `sqrt(a^2-4)tanA+atanB+sqrt(a^2+4)tanC=6a ,` then the least value of `tan^2A+tan^2B+tan^2C` is a. `6` b. `10` c. `12` d. `3`

Options

A. 6
B. 10
C. 12
D. 3

Answer

d

Explanation

The given relation can be rewritten as the vector expression
` (sqrt(a^(2)-4) hati + ahatj + sqrt(a^(2) + 4hatk)`
` (tan A hati tanB hatj + tan C hatk) = 6a`
`( sqrt( a^(2) -4 =a^(2) =a^(2) +4))`
`(sqrt(tan^(2)A + tan^(2) B+ tan^(2) C)). (cos theta) = 6a`
` ( veca. vecb = |veca||vecb|cos theta)`
` sqrt3 a sqrt(tan^(2) A + tan^(2) B + tan^(2)C). (cos theta) = 6a`
`tan^(2) A + tan^(2)B + tan^(2)C = 12 sec^(2) theta ge 12`
` (sec^(2) theta ge 1)`
the least value of `tan^(2) A + tan^(2)B + tan^(2)B+ tan^(2)C is 12`.