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# An equilateral triangle is inscribed in the parabola y^2 = 4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

8.   An equilateral triangle is inscribed in the parabola $y^2 = 4 ax$, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

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Given, an equilateral triangle inscribed in parabola with the equation.$y^2 = 4 ax$

The one coordinate of the triangle is A(0,0).

Now, let the other two coordinates of the triangle are

$B(x,\sqrt{4ax})$ and $C(x,-\sqrt{4ax})$

Now, Since the triangle is equilateral,

$BC=AB=CA$

$2\sqrt{4ax}=\sqrt{(x-0)^2+(\sqrt{4ax}-0)^2}$

$x^2=12ax$

$x=12a$

The coordinates of the points of the equilateral triangle are,

$(0,0),(12,\sqrt{4a\times 12a}),(12,-\sqrt{4a\times 12a})=(0,0),(12,4\sqrt{3}a)\:and\:(12,-4\sqrt{3}a)$

So, the side of the triangle is

$2\sqrt{4ax}=2\times4\sqrt{3}a=8\sqrt{3}a$

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