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JEE MainMathematicsStraight Lines

Let A(8, 4) , B(3 + 5 , 4 + 5 ) and C(3 + 5 , 4 - 5 ) be the vertices of a triangle. As varies, the locus of the orthocenter of the triangle is a circle given by the equation x^2 + y^2 + 2gx + 2fy + c = 0 . The value of (g + f + c) is equal to ______ .

Correct answer

18

Step-by-step solution

The vertices of the triangle are A(8, 4) , B(3 + 5 , 4 + 5 ) and C(3 + 5 , 4 - 5 ) . Let us find the distance of each vertex from the point P(3, 4) : PA = (8 - 3)^2 + (4 - 4)^2 = 5 PB = (3 + 5 - 3)^2 + (4 + 5 - 4)^2 = 5 PC = (3 + 5 - 3)^2 + (4 - 5 - 4)^2 = 5 Since PA = PB = PC = 5 , the circumcenter O of the triangle is (3, 4) . The coordinates of the centroid G(x_G, y_G) are: x_G = 8 + 3 + 5 + 3 + 5 3 = 14 + 5( + ) 3 y_G = 4 + 4 + 5 + 4 - 5 3 = 12 + 5( - ) 3 Let the orthocenter be H(x, y) . The centroid G divides

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