Problem 30 · 2017 Math Kangaroo
Stretch
Geometry & Measurement
Number Theory
divisibilityfactor-pairs
The points A and B lie on a circle with centre M. The point P lies on the straight line through A and M. PB touches the circle in B. The lengths of the segments PA and MB are whole numbers, and PB = PA + 6. How many possible values for MB are there?
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Answer: D — 6
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Hint 1 of 2
PB is tangent, so its square equals the product of the whole secant and its external part (power of the point P).
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Hint 2 of 2
Turn the relation into MB = 6 + 18/PA and require whole numbers.
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Approach: use the tangent-secant power of a point, then count integer solutions
- Power of the point P: \(PB^2 = PA \cdot (PA + 2\,MB)\), since the secant through A and M has external part PA and crosses the circle again a diameter (2·MB) further on.
- With PB = PA + 6: \((PA+6)^2 = PA^2 + 2\,PA\cdot MB\) gives \(12\,PA + 36 = 2\,PA\cdot MB\), so \(MB = 6 + \dfrac{18}{PA}\).
- MB is a whole number when PA divides 18: PA ∈ {1, 2, 3, 6, 9, 18}, giving 6 distinct values of MB, choice D.
Mark:
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