Third Millennium International Mathematical Olympiad 2009
Grade 10-12
Problem 6
In a Cartesian system of coordinates, a circle of radius r with center at (p, q) meets the parabola with the equation |
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Copyright © 1996-2018 Alexander Bogomolny
The equation of a circle with center (p, q) and radius r is (x - p)² + (y - q)² = r². We may eliminate x² by multiplying this equation by a and subtracting the result from the equation of the parabola,
(*) | (2pa + b)x = ay² + (1 - 2aq)y + a (p² + q² - r²). |
Dividing by (2pa + b) we indeed get an equation of a parabola
x = Ay² + By + C, |
where A = a / k, B = (1 - 2aq) / k, C = a (p² + q² - r²) / k, k = 2pa + b. However, let's not forget that division is not always possible:
For that problem, Sadik Shahidain received the score of 5 (out of 7) for the following observation:
There is not necessarily another parabola passing through the same four points. In the case of the circle centered at |
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Copyright © 1996-2018 Alexander Bogomolny
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