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Given nonzero real numbers a, r, s, and t, consider the points P(a², 2at), Q, R(ar², 2ar), and S(as², 2as), all lying on the parabola y² = 4ax. Assume PQ is a focal chord and that the lines QR and PK are parallel, where K is the point (2a, 0). What is the value of r?

1. 1/t
2. (t² + 1)/t
3. 1/t
4. (t² - 1)/t

Correct answer: (t² - 1)/t

Solution

The value of r is determined by the condition that PQ is a focal chord and QR is parallel to PK. Using the geometry of the parabola and the given relationships, r simplifies to (t² - 1)/t.

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