Question

An equilateral triangle \(SAB\) is inscribed in the parabola \(y^2 = 4ax\) having its focus at \(S\). If chord \(AB\) lies towards the left of \(S\), then side length of this triangle is

• A. \(2a(2 - \sqrt{3})\)
• B. \(4a(2 - \sqrt{3})\)
• C. \(a(2 - \sqrt{3})\)
• D. \(8a(2 - \sqrt{3})\)

Hint:

Use parametric form and angle condition.

Answer 1

The Correct Option is B

To find the side length of the equilateral triangle \(SAB\) inscribed in the parabola \(y^2 = 4ax\) with its focus at \(S\), we need to follow these steps:

1. Identify the coordinates of the focus of the parabola \(y^2 = 4ax\). The standard form of a parabola \(y^2 = 4ax\) has its focus at \((a, 0)\).
2. Place the vertex \(A\) of the equilateral triangle at the focus \(S = (a, 0)\). The other two vertices \(B\) and \(C\) lie on the parabola equidistant from \(A\), forming an equilateral triangle.
3. Use the property of an equilateral triangle, i.e., all sides are equal. Let's denote the side length as \(l\). The coordinates of \(B\) and \(C\) can be determined using the condition that triangle \(SAB\) is equilateral.
4. Calculate the distance between \(S = (a, 0)\) and any point \((x_1, y_1)\) on the parabola. Since it is equilateral, the distance \(l\) must also equal the distance between the other vertices of the triangle \(S\).
5. Given that \(AB\) is a chord on the left of \(S\), assume the coordinates of \(A\) and \(B\) such that the triangle sides meet the conditions of an equilateral triangle: \[ l = \sqrt{(x_2 - a)^2 + (y_2 - 0)^2} = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \] \[ y^2 = 4ax \]
6. Since the triangle is equilateral with its sides along the parabola, derive an equation involving the side length \(l\).
7. Use the geometry for an equilateral triangle and properties of parabolas to find: \[l = 4a(2 - \sqrt{3})\] This calculation involves solving the necessary algebra to evaluate the coordinates of the other two vertices \(B\) and \(C\), ensuring the equilateral condition is satisfied.

Thus, the side length of the equilateral triangle \(SAB\) inscribed in the parabola with the chord \(AB\) to the left of \(S\) is 4a(2 - \sqrt{3}).