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JEE MainMathematicsStraight Lines

Let H and P be the orthocenter and circumcenter, respectively, of a triangle with vertices A(0, 12) , B(0, 0) and C(16, 0) . If H' and P' are the images of H and P when reflected in the line 2x + y - 6 = 0 , and the midpoint of the line segment H'P' is ( , ) , then the value of + is :

Options

  1. A4
  2. B1
  3. C13
  4. D7

Correct answer

B. 1

Step-by-step solution

The given triangle has vertices A(0, 12) , B(0, 0) and C(16, 0) . Since the sides AB and BC lie on the coordinate axes, the triangle is right-angled at B . For a right-angled triangle, the orthocenter H is the vertex containing the right angle. Thus, H is (0, 0) . The circumcenter P is the midpoint of the hypotenuse AC . P = ( 0 + 16 2 , 12 + 0 2 ) = (8, 6) . Let M be the midpoint of the line segment HP . M = ( 0 + 8 2 , 0 + 6 2 ) = (4, 3) . Since reflection is a linear transformation, the midpoint of the images H'

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