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JEE MainMathematicsStraight Lines

In an isosceles triangle ABC ( AB = AC ), vertex A is at (9, -6) and the circumcenter O is at (4, -1) . Vertex B lies on the line x - 7y - 11 = 0 and has a negative x -coordinate. If the centroid of the triangle is G( , ) , the value of 3( - ) is equal to

Options

  1. A37
  2. B15
  3. C9
  4. D13

Correct answer

D. 13

Step-by-step solution

The circumradius R is the distance OA . R^2 = OA^2 = (9 - 4)^2 + (-6 - (-1))^2 = 5^2 + (-5)^2 = 50 Since B lies on x - 7y - 11 = 0 , let B be (7y + 11, y) . As O is the circumcenter, OB^2 = R^2 = 50 . (7y + 11 - 4)^2 + (y - (-1))^2 = 50 (7y + 7)^2 + (y + 1)^2 = 50 49(y + 1)^2 + (y + 1)^2 = 50 50(y + 1)^2 = 50 (y + 1)^2 = 1 y + 1 = 1 So, y = 0 or y = -2 . If y = 0 , x = 11 . But the x -coordinate of B must be negative, so this is rejected. If y = -2 , x = -3 . Thus, B (-3, -2) . In an isosceles triangle ( AB = AC ),

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