small and big circles

Geometry Level 2

In the figure shown above, the small circle has its center at P P while the big circle has its center at C C . If P A = 3 , B C = 4 , PA=3,BC=4, and P C = 6 PC=6 , what is the length of A B ? AB?

If your answer is of the form a b , \frac{a}{b}, where a a and b b are coprime positive integers, find a + b . a+b.


The answer is 14.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Let r r be the radius of the big circle. By the power of a point P P with respect to the larger circle, we have

P A × P B = ( P C ) 2 − r 2 PA \times PB=(PC)^2-r^2

3 P B = 6 2 − 4 2 3PB=6^2-4^2

3 P B = 36 − 16 3PB=36-16

3 P B = 20 3PB=20

P B = 20 3 PB=\dfrac{20}{3}

It follows that,

A B = 20 3 − 3 = 20 3 − 9 3 = 11 3 AB=\dfrac{20}{3} - 3=\dfrac{20}{3}-\dfrac{9}{3}=\dfrac{11}{3}

Finally,

a + b = 11 + 3 = a+b=11+3= 14 \boxed{14}

13 Helpful 1 Interesting 0 Brilliant 1 Confused

can someone explain why its pa * pb = pc^2 -r^2 and not just pa * pb = 2^2? Because can't you just let the point directly in front of p and on the bigger circle be D and so pd = pc - cb (cd=cb) and so pd = 2?

Yash Porwal - 1 year, 5 months ago

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(PC)^2 -r^2 is just equal to (PC + r)(PC - r)

Jerry Yu - 7 months, 2 weeks ago

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