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JEE MainMathematicsStraight Lines

Two vertices of a triangle are A(1, 2) and B(5, 5) , and its orthocenter is H(3, 4) . If the third vertex is C( , ) , then the value of - is equal to

Options

  1. A2
  2. B22
  3. C0
  4. D10

Correct answer

B. 22

Step-by-step solution

Let the vertices of the triangle be A(1, 2) , B(5, 5) , and C( , ) . The slope of side AB is m_ AB = 5 - 2 5 - 1 = 3 4 . The altitude from C is perpendicular to AB , so its slope is - 4 3 . Since it passes through the orthocenter H(3, 4) , its equation is: y - 4 = - 4 3 (x - 3) 4x + 3y = 24 (i) The slope of the altitude AH is m_ AH = 4 - 2 3 - 1 = 1 . The side BC is perpendicular to the altitude AH , so its slope is -1 . Since it passes through B(5, 5) , its equation is: y - 5 = -1(x - 5) x + y = 10 (ii) Vertex C i

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