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JEE MainMathematicsStraight Lines

If the points (k, 14) , (-1, k) and (0, 6) are collinear for two distinct values of k , say k₁ and k₂ (where k₁ < k₂ ), then the area of the triangle with vertices (k₁, k₂) , (k₂, k₁) and (2, 2) is

Options

  1. A10
  2. B4
  3. C2
  4. D20

Correct answer

C. 2

Step-by-step solution

Since the points (k, 14) , (-1, k) and (0, 6) are collinear, the slope of the line segment joining the first two points must equal the slope of the line segment joining the last two points. k - 14 -1 - k = 6 - k 0 - (-1) k - 14 -1 - k = 6 - k Cross-multiplying, we get: k - 14 = (6 - k)(-1 - k) k - 14 = -6 - 6k + k + k^2 k^2 - 6k + 8 = 0 Factoring the quadratic equation: (k - 2)(k - 4) = 0 Thus, the two values of k are 2 and 4 . Since k₁ The vertices of the new triangle are (2, 4) , (4, 2) and (2, 2) . The area of t

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