An equilateral triangle is inscribed in the parabola $y^{2} = 4 a x$ , such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $8 a 3$

Detailed Solution

By symmetry the other two vertices are $(h, k)$ and $(h, - k)$ , and each side from the origin makes $30^{\circ}$ with the x-axis, so $k = \frac{h}{3}$ .

$(h, k)$ lies on $y^{2} = 4 a x$ : $\frac{h ^{2}}{3} = 4 ah$ , so $h = 12 a$ and $k = 43 a$ .

Side $= 2 k = 8 a 3$ .

✓ Check

Distance from the origin to

(12 a, 43 a)

144 a^{2} + 48 a^{2} = 192 a = 83 a

, the same as

2 k

⚠ Common trap

(C)

4 a 3

is only

k

, the half-side above the axis.

Quick tip

Vertex at the parabola's vertex: side $= 83 a$ .

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Parabola — Preparation

This question tests your understanding of Parabola, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix), so its eccentricity is exactly 1. The standard form y^2 = 4ax has its vertex at the origin, focus at (a, 0), directrix x = -a, and latus rectum of length 4a. The other standard forms open left, up or down. Exam questions test identifying the focus, directrix, axis and latus rectum, the parametric point (at^2, 2at), focal distances, and equations of tangents and normals. A key result is that the line y = mx + c touches y^2 = 4ax when c = a/m. Drawing a quick sketch of which way the parabola opens prevents most sign errors. Practise the parabola questions below, each with a step-by-step solution.

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An equilateral triangle is inscribed in the parabola $y^{2} = 4 a x$ , such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:

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Parabola — Preparation

An equilateral triangle is inscribed in the parabola y2=4ax, such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is:

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More Parabola Questions

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Parabola — Preparation

An equilateral triangle is inscribed in the parabola $y^{2} = 4 a x$ , such that one of the vertices of the triangle coincides with the vertex of the parabola. The length of the side of the triangle is: