Externally and internally tangent

Geometry Level 3

I have 3 circles whose ratio of radii is $1:2:3$ . I draw these 3 circles such that they are externally tangent to each other. I then draw another circle such that it is internally tangent to all these 3 circles.

Of all the 4 circles drawn, what is the ratio of areas between the largest circle and the smallest circle?

The answer is 36.

1 solution

Sharky Kesa
Jan 4, 2017

Let the radius or the 4th circle be $r$ . By Descartes' Circle Theorem , we have

$\begin{aligned} -\frac{1}{r} &= \frac{1}{1} + \frac{1}{2} + \frac{1}{3} - 2\sqrt{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}\\ &= \frac{11}{6} - 2\\ &=-\frac{1}{6}\\ \implies r&=6\\ \end{aligned}$

Therefore, the ratio between the areas of the of the largest and smallest circles is $\frac{6^2}{1^2}=36$ .

Great!

Challenge Master Note: What would the answer be if I replaced the word "internally" with "externally" in the problem statement?

Pi Han Goh - 4 years, 5 months ago

$r=\dfrac{6}{23}$

so, answer is $\frac{36}{529}$

Answer is wrong:

$\text{Ans. } = \left (3 \div \dfrac{6}{23} \right)^2 = 132 \dfrac{1}{4}$

Sharky Kesa - 4 years, 5 months ago

No, the answer is $\left( 3 \div \dfrac 6{23} \right)^2 = 132 \dfrac14$ .

Pi Han Goh - 4 years, 5 months ago

@Pi Han Goh – Dammit! Biggest circle, not circle with radius 1.

Sharky Kesa - 4 years, 5 months ago

Externally and internally tangent

The answer is 36.

1 solution

0 pending reports