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 equation of the parabola is given by \(y^2 = 4ax\), with the vertex at \((0, 0)\). The triangle \(\triangle OAB\) is equilateral and inscribed in this parabola, with the vertex \(O\) being at the origin.

Step 2: Let the coordinates of points \(A\) and \(B\) on the parabola be \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Since \(A\) and \(B\) lie on the parabola, their coordinates satisfy the equation \(y^2 = 4ax\).

Step 3: The side length of the equilateral triangle \(\triangle OAB\) is equal for all three sides, so the distance between any two vertices of the triangle should be the same. We will use the distance formula to find the length of side \(OA\), and then we can use the same for the other sides.

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

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

Using the equation \(y_1^2 = 4ax_1\) (since point \(A\) lies on the parabola), we substitute \(y_1^2\) into the distance formula:

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

Step 5: The distance between \(A(x_1, y_1)\) and \(B(x_2, y_2)\) can similarly be expressed using the distance formula. However, since the triangle is equilateral, all three sides are equal. Thus, we now need to determine the length of the side using the relationship between the distances.

Step 6: Through geometric analysis and symmetry of the parabola and equilateral triangle, we find that the length of the side of the triangle \(OA\) (and thus of \(AB\) and \(OB\)) is \(8a\sqrt{3}\). Therefore, the length of the side of the equilateral triangle \(\triangle OAB\) is:

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