Question

A rectangle $R$ with end points of the one of its dies as $(1,2)$ and $(3,6)$ is inscribed in a circle. If the equation of a diameter of the circle is $2x-y+4=0$ , then the area of $R$ is____.

Answer 1

Answer: 16

Eq. of line $AB$
$y=2x$
Slope of $AB=2$
Slope of given diameter $=2$
So the diameter is parallel to $AB$
Distance between diameter and line $AB$
$=\left(\frac{4}{\sqrt{2^2+12}}\right)=\frac{4}{\sqrt{5}}$
Thus $BC=2\times \frac{4}{\sqrt{5}}=\frac{8}{\sqrt{5}}$
$AB=\sqrt{(1-3{)}^2+(2-6{)}^2}=\sqrt{20}=2\sqrt{5}$
Area $=AB\times BC=\frac{8}{\sqrt{5}}\times 2\sqrt{5}=16$