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A parabola has its focus at the point (2, 3) and its tangent at the vertex is given by the equation 3x + 4y + k = 0 . If the length of its latus rectum is 12 , then the absolute difference between the possible values of k is

Options

  1. A120
  2. B60
  3. C6
  4. D30

Correct answer

D. 30

Step-by-step solution

The length of the latus rectum of a parabola is 4a , where a is the perpendicular distance from the focus to the tangent at the vertex. Given that the length of the latus rectum is 12 , we have: 4a = 12 a = 3 The perpendicular distance from the focus (2, 3) to the tangent at the vertex 3x + 4y + k = 0 is equal to a . Using the distance formula: |3(2) + 4(3) + k| 3^2 + 4^2 = 3 |6 + 12 + k| 5 = 3 |18 + k| = 15 This gives two possible equations: 18 + k = 15 k = -3 18 + k = -15 k = -33 The absolute difference between t

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