Question

Area of the hexagon formed by points of contact of direct common tangents to the circle $(x-1{)}^2+(y-3{)}^2=9$, $(x-7{)}^2+(y+5{)}^2=25$ with the two centres is

• A. $25\sqrt{6}\text{sq. units}$
• B. $30\sqrt{6}\text{sq. units}$
• C. $45\sqrt{6}\text{sq. units}$
• D. $32\sqrt{6}\text{sq. units}$

Answer 1

The correct option is D $32\sqrt{6}\text{sq. units}$

Given circles are
$(x-1{)}^2+(y-3{)}^2=9$, $(x-7{)}^2+(y+5{)}^2=25$
The centre and radius are
$C_1\equiv (1,3)r_1=3$
$C_2\equiv (7,-5)r_2=5$


$d=C_1{C}_2=\sqrt{6^2+8^2}=10$
Length of D.C.T
$AB=CD=\sqrt{d^2-(r_2-r_1{)}^2}=\sqrt{100-4}=4\sqrt{6}$

Area of hexagon $A{C}_{1}CD{C}_{2}B$
$=2\times \text{area of trapezium}A{C}_1{C}_{2}B=2\times \frac{1}{2}(3+5)\times \left(4\sqrt{6}\right)=32\sqrt{6}\text{sq. units}$