Question

The ratio of area of incircle and circumcircle of quadrilateral formed by lines $x=1,x=5,y=-1,y=3$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

The correct option is A $\frac{1}{2}$


Centre is the midpoint of $(5,3)$ and $(1,-1)$
$c=(\frac{1+5}{2},\frac{-1+3}{2})\equiv (3,1)$

$r_{\text{incircle}}=\frac{|y_2-y_1|}{2}=\frac{3+1}{2}=2r_{\text{circumcircle}}=\text{distance between (5,3) \&(3,1)}\Rightarrow r_{\text{circumcircle}}=\sqrt{4+4}=2\sqrt{2}\therefore \frac{\text{area of incircle}}{\text{area of circumcircle}}=\frac{2^2}{(2\sqrt{2}{)}^2}=\frac{1}{2}$