Problem 25 · 2019 Math Kangaroo
Stretch
Algebra & Patterns
substitutionfactorization
Four different straight lines pass through the origin of the coordinate system. They intersect the parabola \(y = x^{2} - 2\) at eight points. What could the product of the \(x\)-coordinates of these eight points be?
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Answer: A — only 16
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Hint 1 of 2
A line \(y = mx\) through the origin meets \(y = x^{2} - 2\) where \(x^{2} - mx - 2 = 0\); by Vieta the two \(x\)-roots multiply to \(-2\).
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Hint 2 of 2
Each of the four lines contributes a root-product of \(-2\), so just multiply across the four lines.
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Approach: use Vieta on each line–parabola intersection
- A line \(y = mx\) meets \(y = x^{2} - 2\) where \(x^{2} - mx - 2 = 0\), whose two roots multiply to \(-2\) (the constant term).
- The four lines give four such pairs, each with root-product \(-2\).
- So the product of all eight \(x\)-coordinates is \((-2)^{4} = 16\), no matter which four lines are chosen.
- Answer (A) only 16.
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