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Question
An equilateral triangle is inscribed in the parabola y2 = 4ax whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
Solution
As shown in the figure APQ denotes the equilateral triangle with its equal sides of length l (say).
Here AP = l
So AR = l cos30°
= `l sqrt(3)/2`
Also, PR = `l sin 30^circ = l/2`.
Thus `(lsqrt(3))/2, l/2` are the coordinates of the point P lying on the parabola y2 = 4ax.
Therefore, `l^2/4 = 4a (lsqrt(3))/2`
⇒ `l = 8 asqrt(3)`.
THus, 8 `asqrt(3)` is the required length of the side of the equilateral triangle inscribed in the parabola y2 = 4ax.
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