Question

$2x-y+4=0$ is the diameter of circle passes through the vertices of rect. ABCD $A(4,6)B(1,9)$.Find the area of the rect. ABCD.

Answer 1

Clearly the Given points does not satisfy the given equation

So, The Diameter must pass through C,D

Now, the Rectangle ABCD is a combination of Equal Triangles $BCD,ACD$

So, Area of Rectangle $=2\left(Areaof△BCD\right)=2\ast \frac{1}{2}\ast OB\ast CD$

Now, OB$=\left|\frac{2\left(1\right)-9+4}{\sqrt{2^2+1^2}}\right|=\frac{3}{\sqrt{5}}$

and $CD=AB=\sqrt{3^2+3^2}=3\sqrt{2}$

Hence, Area = $\frac{3}{\sqrt{5}}\ast 3\sqrt{2}=9\sqrt{\frac{2}{5}}$