In a `Delta ABC " if " |(1,a,b),(1,c,a),(1,b,c)| =0`, then `sin^(2) A + sin^(2) B + sin^(2) C` is

Options

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

Answer

B

Explanation

Given, `|{:(1,a,b),(1,c,a),(1,b,c):}|=0`
`impliesc^(2)-ab-a(c-a)+b(b-c)=0`
`impliesa^(2)+b^(2)+c^(2)-ab-bc-ca=0`
`implies1/2[2a^(2)+2b^(2)+2x^(2)-2ab-2bc-2ca=0`
`implies1/2[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]=0`
`impliesa=b=c`
`impliesangleA=60^(@), angleB=60^(@),`
`angleC=60^(@)`
`thereforesin^(2)A+sin^(2)B+sin^(2)C`
`=sin^(2)60^(@)+sin^(2)60^(@)+sin^(2)60^(@)`
`=((sqrt3)/(2))+((sqrt3)/(2))^(2)+((sqrt3)/(2))=9/4`