An equilateral triangle ABC is inscribed in the parabola y = x² and one of the side of the equilateral triangle has the gradient 2. If the sum of x-coordinates of the vertices of the triangle is a rational in the form $\frac{p}{q}$ where p and q are coprime, then the value of $\frac{(p + q)}{7}$ is

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answer is 2.

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Detailed Solution

$A = (t_{1}, t_{1}^{2}), B = (t_{2}, t_{2}^{2}), C = (t_{3}, t_{3}^{2})$
$\Rightarrow$ Slope of $AB = t_{1} + t_{2} = 2 (∵ data) = \tan θ$
Slope of $BC = t_{2} + t_{3} \equiv \tan (θ + 60^{°})$
$\Rightarrow$ Slope of $A C = t_{1} + t_{3} = \tan (θ - 60^{\circ})$
( $∴$ equalateral)
Adding, $2 (t_{1} + t_{2} + t_{3}) = 2 + \frac{2 + \sqrt{3}}{1 - 2 \sqrt{3}} + \frac{2 - \sqrt{3}}{1 + 2 \sqrt{3}} = 2 - \frac{16}{11} = \frac{6}{11}$
$\Rightarrow \frac{p}{q} = t_{1} + t_{2} + t_{3} = \frac{3}{11} \Rightarrow \frac{p + q}{7} = \frac{3 + 11}{7} = 2$