Question

\(\triangle OAB\) is an equilateral triangle inscribed in the parabola \(y^2 = 4ax, \, a>0\) with \(O\) as the vertex. Then the length of the side of \(\triangle OAB\) is:

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

Hint:

For equilateral triangles inscribed in conic sections like parabolas, use symmetry and the distance formula to calculate the side lengths and properties of the triangle.

Answer 1

The Correct Option is A

Step 1: The parabola's equation is \(y^2 = 4ax\), with vertex at \((0, 0)\). The equilateral triangle \(\triangle OAB\) is inscribed within this parabola, with vertex \(O\) at the origin.

Step 2: Points \(A\) and \(B\) on the parabola have coordinates \((x_1, y_1)\) and \((x_2, y_2)\), respectively. These points satisfy the equation \(y^2 = 4ax\).

Step 3: In equilateral triangle \(\triangle OAB\), all sides are equal. We'll use the distance formula to find side \(OA\).

Step 4: The distance between origin \(O(0, 0)\) and point \(A(x_1, y_1)\) is:

\[ OA = \sqrt{x_1^2 + y_1^2} \]

Using \(y_1^2 = 4ax_1\), we substitute into the distance formula:

\[ OA = \sqrt{x_1^2 + 4ax_1} \]

Step 5: The distance \(AB\) can also be found using the distance formula. Since the triangle is equilateral, all sides are equal, and we must determine the side length using the distance relationships.

Step 6: Geometric analysis and symmetry show side \(OA\) (and thus \(AB\) and \(OB\)) is \(8a\sqrt{3}\). Therefore, the equilateral triangle \(\triangle OAB\) side length is:

\[ 8a\sqrt{3} \text{ units.} \]