Question bank/JEE Main/Mathematics

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and
the angle between them is 60º. If the area of the quadrilateral is
$4\sqrt{3}$, then the perimeter of the quadrilateral is

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$\frac{x^{2} + y^{2} - 19}{2xy} = - \frac{1}{2}$
$BD = \sqrt{19}$ x^2 + y^2 - 19 = - xy....... (1)
$\frac{1}{2} \times 10 \times \frac{\sqrt{3}}{2} + \frac{1}{2} \times xy\frac{\sqrt{3}}{2} = 4\sqrt{3}$
xy = 6...... (2)
from (1) and (2)
(x, y) = (3, 2) or (2, 3)
Perimeter = 3 + 2 + 5 + 2 = 12

Source: Grass-G / M-24 Mathematics Paper 31 Dec.docx