Question

If $2x-y+4=0$ is a diameter of a circle which circumscribes a rectangle ABCD. If the co-ordinates of $A,B$ are $(4,6)$ and $(1,9)$ , find the area of ABCD , and show that it is a square.

Answer 1

None of the points A or B satisfy the equation of diameter and hence given diameter is not along the diagonals AC or BD . If the centre be (h,k), then $2h-k+4=0$ ....(1)

If Q($\frac{5}{2}$, $\frac{15}{2}$) be mid-point of AB , then PQ is perpendicular to AB

$\therefore m_1{m}_2=-1givesk=h+5$ ......(2)

Solving (1) and (2) , we get $P(1,6)$

Now $AB=3\sqrt{2},PQ=3\sqrt{2}$

$\therefore AD=2PQ=3\sqrt{2}$

$\therefore$ Rectangle is a square of side $3\sqrt{2}$ so that its area $3(\sqrt{2}{)}^2=18$