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PLEASE HELP!! If the line y=root 3x interests the curve x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0 At 3 points A,B,C .Then find OA*OB*OC...
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y=root 3x ..............i x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0 ............ii Now, From i and ii, (1+3root3)x^3 + (14+3root3)x^2 + (4+5root3)x-1=0 x^3 + (14+3root3)/(1+3root3)x^2 + (4+5root3)/(1+3root3)x -1/(1 + 3root3) =0 Now product of all the roots of this cubic equation is -1/(1+3root3) Now, OA*OB*OC = 8*3root3*product of all the roots of the above cubic equation =-24root3 /(1+3root3)
\[x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0\] is the equation :(
\[y=\sqrt{3}x\] is the line
oh yes it is 8/(3root3 +1) y=root 3x ..............i x^3+y^3+3xy+5x^2+3y^2+4x+5y-1=0 ............ii Now, From i and ii, (1+3root3)x^3 + (14+3root3)x^2 + (4+5root3)x-1=0 x^3 + (14+3root3)/(1+3root3)x^2 + (4+5root3)/(1+3root3)x -1/(1 + 3root3) =0 Now product of all the roots of this cubic equation is -{-1/(1+3root3)} NOTE: it is -ve since it is a cubic equation which has odd number of roots Now, OA*OB*OC = 8*product of all the roots of the above cubic equation =8/(1+3root3)