Question

Two circles I and II are externally tangent. A tangent to the circle I passes through the centre of the circle II. The distance from the point of tangency to the centre of the circle II is three times the radius of the circle II. The ratio of the circumference of the circle I to the circumference of the circle II

• A. $2$
• B. $3$
• C. $4$
• D. $16$

Answer 1

Answer: (C)

$4$

From the above figure,O'RO is a right triangle.
By Pythagoras theorem,
${OP}^2+{P{O}^{\prime }}^2={OO}^{\prime 2}$
$⟹{\left(3a\right)}^2+r^2={(a+r)}^2⟹9a^2+r^2=a^2+r^2+2ar⟹9a^2=a^2+2ar⟹8a^2=2arorr=4a$

$⟹\frac{r}{a}=4$.