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Find a subset of the complete set of values of slope m for which a chord of the circle x² + y² = 4 having slope m can be tangent to the parabola y² = 4x. Which interval below is contained in the valid set?
- (-infinity, -sqrt((sqrt(2)-1)/2))
- (0, 1/2)
- (sqrt((sqrt(2)-1)/2), infinity)
- (-1/2, 0)
Correct answer: (-infinity, -sqrt((sqrt(2)-1)/2))
Solution
Imposing both tangency to the parabola and the chord (intersection) condition on the circle yields an inequality in m² whose solution set includes the large-|m| intervals, e.g. m < -sqrt((sqrt(2)-1)/2).
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