A rectangle is inscribed in a circle with a diameter lying along...

Given, the two adjacent vertices of the rectangle are $\left(- 8, 5\right)$ and \left(6, 5\right),$the$y$-co-ordinates of both of these points is same this means they lie on the line$y = 5,$which is parallel to the$x$-axis.</p><p>And, we know that the sides of a rectangle are perpendicular to each other, hence the other two sides passing through these vertices are parallel to$y$-axis.</p><p>Also, we know that the equation of a line parallel$y$-axis is$x = k .$</p><p>Therefore, the equation of the sides of the rectangle passing through \left(- 8, 5\right)$ and \left(6, 5\right)$are respectively$x = - 8$and$x = 6 .$</p><p>And, hence the other two vertices of the rectangle can be taken as \left(6, a\right)$ and \left(- 8, b\right) .$</p><p>And since the other side is parallel to$x$-axis, hence$a = b .$</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/df01fcd9-ff1f-4685-956d-b9b82284be2f-image2.png" style="width: 250px; height: 260px;" /></p><p>Here, diagonal subtends right angle at the circumference of circle, so its mid-point is the same as centre of circle, which lies on the given diameter.</p><p>The mid-point of a line segment joining the points$\left({x}_{1}, {y}_{1}\right)$and \left({x}_{2}, {y}_{2}\right)$ is \left(\dfrac{{x}_{1} + {x}_{2}}{2}, \dfrac{{y}_{1} + {y}_{2}}{2}\right),$hence, the mid-point of the diagonal of the rectangle is \left(\dfrac{- 8 + 6}{2}, \dfrac{a + 5}{2}\right) = \left(- 1, \dfrac{a + 5}{2}\right)$

This point lie on the diameter $3y = x + 7$

So, $\dfrac{3\left(a + 5\right)}{2} = - 1 + 7$
$⇒ a + 5 = \dfrac{2\left(6\right)}{3}$

$⇒ a = - 1$

Hence, the vertices are \left(- 8, 5\right), \left(6, 5\right), \left(6, - 1\right)$and \left(- 8, - 1\right) .$

The distance between the points \left({x}_{1}, {y}_{1}\right)$and \left({x}_{2}, {y}_{2}\right)$ is $\sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}}$

Thus, the length of sides are $6$ and $14$ units.

Hence, area $= l \times b = 14 \times 6 = 84$ sq. units