Question

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:

• A. \( (2, 0) \)
• B. \( (\sqrt{2}, 0) \)
• C. \( (1, 0) \)
• D. \( (\sqrt{3}, 0) \)

Hint:

For an equilateral triangle inscribed in a parabola, the centroid lies along the axis of symmetry of the parabola.

Answer 1

The Correct Option is A

We are given that an equilateral triangle is inscribed in the parabola \( y^2 = x \), and one vertex of the triangle is at the vertex of the parabola, which is at \( (0, 0) \). The equation of the parabola is \( y^2 = x \), and the vertex of the parabola is at \( (0, 0) \). Since the triangle is equilateral, the other two vertices of the triangle must lie symmetrically on the parabola. The vertices of the triangle are at \( (0, 0) \) and two other points on the parabola. To find the centroid, we use the fact that the centroid of an equilateral triangle divides each median into a ratio of 2:1. By calculating the coordinates of the other vertices and the centroid, we find that the centroid of the triangle is at \( (2, 0) \).