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Question

If the line $y equals x square root of 3$ cuts the curve $x to the power of 3 end exponent plus y to the power of 3 end exponent plus 3 x y plus 5 x to the power of 2 end exponent plus 3 y to the power of 2 end exponent plus 4 x plus 5 y minus 1 equals 0$ at the points A,B and $C$ then OA.OB.OC =-( where O is origin)

$\frac{12 \sqrt{3} - 4}{13}$
$\frac{3 \sqrt{3} + 12}{13}$
$\frac{4 (3 \sqrt{13} - 1)}{13}$
$\frac{4 \sqrt{3} - 3}{13}$

The correct answer is: $fraction numerator 12 square root of 3 minus 4 over denominator 13 end fraction$

Using symmetric form of straight line $fraction numerator x over denominator blank c o s blank 6 0 to the power of o end exponent end fraction equals fraction numerator y over denominator blank s i n blank 6 0 to the power of o end exponent end fraction equals r$ cuts given curve
$left parenthesis fraction numerator r over denominator 2 end fraction right parenthesis to the power of 3 end exponent plus left parenthesis to the power of fraction numerator gamma square root of 3 over denominator 2 end fraction end exponent right parenthesis left parenthesis right parenthesis to the power of 3 end exponent plus 3 left parenthesis fraction numerator r over denominator 2 end fraction right parenthesis subscript left parenthesis to the power of fraction numerator gamma square root of 3 over denominator 2 end fraction end exponent right parenthesis end subscript superscript left parenthesis right parenthesis end superscript plus 5 left parenthesis fraction numerator r over denominator 2 end fraction right parenthesis to the power of 2 end exponent plus 3 left parenthesis to the power of fraction numerator gamma square root of 3 over denominator 2 end fraction end exponent right parenthesis left parenthesis right parenthesis to the power of 2 end exponent$
$plus 4 left parenthesis fraction numerator r over denominator 2 end fraction right parenthesis plus 5 left parenthesis to the power of fraction numerator r square root of 3 over denominator 2 end fraction end exponent right parenthesis to the power of negative 1 end exponent left parenthesis right parenthesis equals 0$
$Error converting from MathML to accessible text.$
$Error converting from MathML to accessible text.$
is a cubic equation whose product of roots is OA.OB.OC $Error converting from MathML to accessible text.$

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