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Let B and C be two points on the line 4x + 3y = 0 such that B and C are symmetric with respect to the origin. Suppose A is a point in the first quadrant on the line 2x + y = k such that ABC is an equilateral triangle. If the area of ABC is 75 3 , then the value of k is

Options

  1. A33
  2. B30
  3. C22 3
  4. D-33

Correct answer

A. 33

Step-by-step solution

Since B and C lie on 4x + 3y = 0 and are symmetric with respect to the origin, the origin (0,0) is the midpoint of BC . In an equilateral triangle, the median is also the altitude. Thus, the altitude from A to BC lies on the perpendicular bisector of BC , which passes through the origin. The equation of the perpendicular bisector is 3x - 4y = 0 . Let the coordinates of A be (4t, 3t) . The length of the altitude h is the distance from A to the origin: h = (4t)^2 + (3t)^2 = 5|t| . The area of an equilateral triangle

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