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A parabola has its focus at F(1, 1) and is tangent to the line x + y = 7 at the point P(3, 4). Which of the following statements about the parabola are correct?

1. Equation of its directrix is 3x + 2y = 30
2. Length of its latus rectum is 50 / sqrt(13)
3. Its vertex is at (6, 6)
4. Its axis has slope 2/3

Correct answer: Equation of its directrix is 3x + 2y = 30

Solution

Reflecting F(1,1) in the tangent x+y=7 gives P'=(6,6) on the directrix. The parabola's axis direction is (3,2)/sqrt(13) (found by requiring FP=sqrt(13) equals the perpendicular distance from P to the directrix). The directrix passes through (6,6) with normal (3,2): 3x+2y=30. Latus rectum = 4a = 2*dist(F,directrix) = 2*5|cos(theta)+sin(theta)| = 50/sqrt(13). The vertex is at (101/26, 38/13), not (6,6).

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