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  4. An equilateral triangle is inscribed in the parabola y2 = 4x one of whose vertex is at the vertex of the parabola, the length of each side of the triangle is

Q. An equilateral triangle is inscribed in the parabola y$^2$ = 4x one of whose vertex is at the vertex of the parabola, the length of each side of the triangle is

VITEEEVITEEE 2006 Report Error

Solution:

Let AB = $\ell , \, then \, \, AM \, = \ell cos 30^{°} \, = \frac{\ell \sqrt{3}}{2}$
& BM = $\ell \, sin 30^{°} \, = \frac{\ell}{2}$
image
So, the coordinates of B are $\bigg( \frac{\ell \sqrt{3}}{2}, \frac{\ell}{2}\bigg)$
Since, B lies on $y^2 \, = \, 4x$
$\therefore \, \, \frac{\ell^2}{4}=4\bigg(\frac{\ell \sqrt{3}}{2}\bigg)$
$\Rightarrow \, \, \ell^2 \, \frac{16}{2}. \sqrt{3\ell} \, \, \Rightarrow \, \, =8\sqrt{3}$

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