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A parabola has its axis along the x-axis and its focus at the point (3, 0) . If the line x - 2y + 7 = 0 is a tangent to this parabola, then the length of its latus rectum is

Options

  1. A8
  2. B8 5
  3. C4 5
  4. D4

Correct answer

A. 8

Step-by-step solution

A key property of a parabola is that the foot of the perpendicular drawn from the focus to any tangent lies on the tangent at the vertex. Let us find the foot of the perpendicular from the focus (3, 0) to the given tangent x - 2y + 7 = 0 . The slope of the tangent is 1 2 . The line perpendicular to it passing through the focus will have a slope of -2 . The equation of this perpendicular line is: y - 0 = -2(x - 3) 2x + y = 6 Now, we find the intersection of the tangent and this perpendicular line: x - 2y = -7 2x + y

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