An equilateral triangle is inscribed in a parabola x^{2}=\sqrt{3}y where one vertex of the triangle is at the vertex of the parabola. If p is the length of side of the triangle and q is the length of the latus rectum, then which one of the following is correct?

Explanation

The parabola is x^2 = \sqrt{3}y, so its latus rectum is q = \sqrt{3}. Let the vertices of the equilateral triangle be (0,0) and (\pm \frac{p}{2}, \frac{p\sqrt{3}}{2}). Substituting the coordinates of one vertex into the parabola's equation gives (\frac{p}{2})^2 = \sqrt{3}(\frac{p\sqrt{3}}{2}) \implies \frac{p^2}{4} = \frac{3p}{2}, which gives p = 6. Since q = \sqrt{3}, we have p = 2\sqrt{3}(\sqrt{3}) = 2\sqrt{3}q.

Related questions on Analytical Geometry (2D)

• Consider the following statements in respect of the line passing through origin and inclining at an angle of 75^{\circ} with the positive ...
• If P(3,4) is the mid-point of a line segment between the axes, then what is the equation of the line ?
• The base AB of an equilateral triangle ABC with side 8 cm lies along the y-axis such that the mid-point of AB is at the origin and B...
• The centre of the circle passing through origin and making positive intercepts 4 and 6 on the coordinate axes, lies on the line
• The centre of an ellipse is at (0,0), major axis is on the y-axis. If the ellipse passes through (3,2) and (1,6), then what is its ecc...