The area (in sq. units) of an equilateral triangle inscribed in the parabola

Question:

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2}=8 x$, with one of its vertices on the vertex of this parabola, is:

  1. $64 \sqrt{3}$

  2. $256 \sqrt{3}$

  3. $192 \sqrt{3}$

  4. $128 \sqrt{3}$


Correct Option: , 3

Solution:

$\tan 30^{\circ}=\frac{4 \mathrm{t}}{2 \mathrm{t}^{2}}=\frac{2}{\mathrm{t}} \Rightarrow \mathrm{t}=2 \sqrt{3}$

$\mathrm{AB}=8 \mathrm{t}=16 \sqrt{3}$

Area $=256.3 \cdot \frac{\sqrt{3}}{4}=192 \sqrt{3}$

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