if the line `y=xsqrt3` cuts the curve `x^3+y^3+3xy+5x^2+3y^2+4x+5y+1=0` at point `A,B,C` then find the value of `OA.OB.OC` is equal to (where `O` is the origin)

Explanation

Let `A(x_(1),y_(1)),B(x_(2),y_(2))and C(x_(3),y_(3))` be the points of intersection of the line`y=sqrt3x` with the curve. Solve curve and line
`x^(3)+3sqrt3x^(3)+3sqrt3x^(2)+5x^(2)+9x^(2)+4x+5sqrt3x-1=0`
`impliesx^(3)(1+3sqrt3)+(14+3sqrt3)x^(2)+(4+5sqrt3)x-1=0" "...(i)`
Now, `OA,OB.OC=sqrt(x_(1)^(2)+y_(1)^(2))sqrt(x_(2)^(2)+y_(2)^(2))sqrt(x_(3)^(2)+y_(3)^(2))`
`andy_(i)=sqrt3x_(i)" "(i=1,2,3)`
`OA.OB.OC=(2x_(1))(2x_(2))(2x_(3))=8x_(1)x_(2)x_(3)`
From equation (i)
`x_(1)x_(2)x_(3)=(1)/(1+3sqrt3)`
`thereforeOA.OB.OC=(8)/(1+3sqrt3)=(3)/(13)(3sqrt3-1)`