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Q.
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 $2 x - y +4= 0$ , then the area of $R$ is____.
Eq. of line $A B$
$y=2 x$
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 $= AB \times BC =\frac{8}{\sqrt{5}} \times 2 \sqrt{5}=16$
