Question

One of the diameters of the circle circumscribing the rectangle $ABCD4y=x+7$. If $A$ and $B$ are the points $(-3,4)$ and $(5,4)$ respectively, then find the area of rectangle.

Answer 1

Let $O$ be the centre of circle and $M$ be mid-point of $AB$.

Then, $OM\perp AB\Rightarrow M(1,4)$
Since, slope of $AB=0$
Equation of straight line $MO$ is $x=1$ and equation of diameter is $4y=x+7$.
$\Rightarrow$ Centre is $(1,2).$
Also, $O$ is mid-point of $BD$
$\Rightarrow (\frac{\alpha +5}{2},\frac{\beta +4}{2})=(1,2)\Rightarrow \alpha =-3,\beta =0$
$\therefore AD=\sqrt{(-3+3{)}^2+(4-0{)}^2}=4$
and $AB=\sqrt{64+0}=8$
Thus, area of rectangle $=8\times 4=32$ sq units