Question

Consider two circles in a plane with radii 5 and 3 respectively. Distance between centres of these two circles is 10 . If $R_1$ is radius of circle passing through 4 points of contact of direct common tangents of these two circles and $R_2$ is radius of circle passing through 4 points of contact of transverse common tangents then $\frac{{\mathrm{R}}_1}{{\mathrm{R}}_2}=$ ?

• A. $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$
• B. 2
• C. $\frac{5}{3}$
• D. $\frac{25}{9}$

Answer 1

Answer: (B)

$\mathrm{FG}=\frac{{\mathrm{r}}_1+r_2}{2}$
$\mathrm{GB}=\sqrt{(FG{)}^2+(FB{)}^2}$
$(\mathrm{GB}{)}^2=\frac{{\left(r_1+r_2\right)}^2}{4}+\frac{d^2-{\left(r_1-r_2\right)}^2}{4}$
$=\frac{d^2+4r_1{r}_2}{4}$
$R_1^2=\frac{{\left(C_1{C}_2\right)}^2+4r_1{r}_1}{4}=40$
$R_2^2=\frac{{\left(C_1{C}_2\right)}^2-4r_1{r}_1}{4}=10$