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

Step 1: Formula / Definition}
\[ S(a, 0), A(at_1^2, 2at_1), B(at_2^2, 2at_2) \]
Step 2: Calculation / Simplification}
\(SA = SB\) and \(\angle ASB = 60^\circ\)
Slope of \(SA\): \(\frac{2at_1}{at_1^2 - a} = \frac{2t_1}{t_1^2 - 1} = \tan 150^\circ = -\frac{1}{\sqrt{3}}\)
\(2\sqrt{3}t_1 = -t_1^2 + 1 \Rightarrow t_1^2 + 2\sqrt{3}t_1 - 1 = 0\)
\(t_1 = -\sqrt{3} \pm 2\). Since left of \(S\), \(t_1 = 2 - \sqrt{3}\)
Side length \(= SA = a(t_1^2 + 1) = a((2-\sqrt{3})^2 + 1) = a(4 - 4\sqrt{3} + 3 + 1) = 4a(2-\sqrt{3})\)
Step 3: Final Answer
\[ 4a(2 - \sqrt{3}) \]