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Three lines are given: the y-axis, the line y = 2, and the line lx + my = 1 where the point (l, m) lies on y² = 4x. The variable line lx + my = 1 is known to be tangent to a fixed parabola. A general point on that parabola can be written in parametric form as which of the following?
- (2 + t²/32, 3/2 + t/16)
- (2 + t²/32, -3/2 + t/16)
- (-2 + t²/32, 3/2 + t/16)
- (-2 + t²/16, 3/2 + t/5)
Correct answer: (2 + t²/32, 3/2 + t/16)
Solution
The envelope of the line family is a parabola; writing its points in vertex-based parametric form matches (2 + t²/32, 3/2 + t/16).
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