Question

Equations of four circles are $(x\pm 5{)}^2+(y\pm 5{)}^2=25.$ Then which of the following is (are) CORRECT?

• A. The minimum radius of the circle touching all the four circles is $5(\sqrt{2}-1)$
  
• B. The maximum radius of circle touching all the four circles is $5(\sqrt{2}+1)$
  
• C. Area of region enclosed by all the four circles is $100-25\pi \text{sq. unit}$
  
• D. The centres of four circles are the vertices of a rectangle

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A The minimum radius of the circle touching all the four circles is $5(\sqrt{2}-1)$

B The maximum radius of circle touching all the four circles is $5(\sqrt{2}+1)$

C Area of region enclosed by all the four circles is $100-25\pi \text{sq. unit}$

D The centres of four circles are the vertices of a rectangle


There are two circles that touch all the four circles as shown in diagram.
In $\Delta AOE,$
$OA=\sqrt{5^2+5^2}=5\sqrt{2}$
$\therefore$ Radius of smaller circle $=5\sqrt{2}-5=5(\sqrt{2}-1)$
Radius of larger circle $=5\sqrt{2}+5=5(\sqrt{2}+1)$

Area enclosed by circles $={10}^2-4\left(\frac{\pi \times 5^2}{4}\right)$
$=100-25\pi \text{sq. unit}$

$AB=BC=CD=DA$
$\Rightarrow ABCD$ is a square.