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JEE MainMathematicsStraight Lines

Two sides of a triangle lie on the lines 2x - y = 0 and x + y = 3 . If the centroid of the triangle is ( 8 3 , 7 3 ) , then the equation of the third side is

Options

  1. A7x + y = 21
  2. B7x + y = 45
  3. C7x + y = 27
  4. Dx + 7y = 45

Correct answer

C. 7x + y = 27

Step-by-step solution

Let the vertices of the triangle be A , B , and C . The coordinates of vertex A can be found by solving the equations of the given sides 2x - y = 0 and x + y = 3 . Substituting y = 2x into the second equation gives x + 2x = 3 x = 1 . Thus, y = 2 , and the vertex is A(1, 2) . Let M(x_m, y_m) be the midpoint of the third side BC . The centroid G ( 8 3 , 7 3 ) divides the median AM in the ratio 2:1 . Using the section formula: x_G = 2x_m + x_A 3 8 3 = 2x_m + 1 3 2x_m = 7 x_m = 7 2 y_G = 2y_m + y_A 3 7 3 = 2y_m + 2 3 2

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