Let `a , ba n dc` detnote the sides `B C ,C Aa n dA B` respectively of ` A B Cdot` If `|1a b1c a1b c|=0`

Options

A. `(9)/(4)`
B. `(4)/(9)`
C. `(3 sqrt3)/(2)`
D. 1

Answer

A

Explanation

We have,
`|(1,a,b),(1,c,a),(1,b,c)| =0`
`rArr a^(2) + b^(2) + c^(2) - ab - bc - ca = 0`
`rArr (1)/(2) {(a -b)^(2) + (b-c)^(2) + (c -a)^(2)} = 0`
`rArr a - b = 0, b -c =0, c -a = 0 rArr a = b = c`
`rArr Delta ABC` is equilateral
`rArr A = B = C = pi//3`
`:. sin^(2) A + sin^(2) B + sin^(2) C = (3)/(4) + (3)/(4) + (3)/(4) = (9)/(4)`