An equilateral triangle is inscribed in the parabola $y^{2} = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is

Answer (Detailed Solution Below)View answer ▾

Answer: Option D — $(2, 0)$

Detailed Solution

By symmetry, the other two vertices are $(t^{2}, t)$ and $(t^{2}, - t)$ on $y^{2} = x$ .

Each side through the origin makes $30^{\circ}$ with the axis: $\frac{t}{t ^{2}} = t an 30^{\circ} = \frac{1}{3}$ , so $t = 3$ and the vertices are $(3, \pm 3)$ .

Centroid $= (\frac{0 + 3 + 3}{3}, \frac{0 + 3 - 3}{3}) = (2, 0)$ .

✓ Check

Sides from the origin are

9 + 3 = 12

and the third side is

23 = 12

: all equal.

⚠ Common trap

(C)

(3, 0)

uses the parameter

t = 3

as the x-coordinate of the centroid.

Quick tip

$\frac{t}{t ^{2}} = tan 3 0^{\circ} \Rightarrow t = 3, G = (\frac{2 t ^{2}}{3}, 0) = (2, 0)$

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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 $y^{2} = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is

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An equilateral triangle is inscribed in the parabola y2=x. One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is

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An equilateral triangle is inscribed in the parabola $y^{2} = x$ . One vertex of the triangle is at the vertex of the parabola. The centroid of the triangle is