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Let P: y² = 4ax (a > 0) be a parabola with focus S. Two tangents to P, each making an angle of pi/4 with the line y = 3x + 5, touch the parabola at A and B. For what value(s) of a are A, B and S collinear?

1. a > 0 only
2. 8 only
3. 2 only
4. 1/4 only

Correct answer: a > 0 only

Solution

The tangent slopes are -2 and 1/2, whose product is -1, so the tangents are perpendicular; their chord of contact is then a focal chord for every a > 0, making A, B, S collinear for all a > 0.

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