Hey I'm the circumcentre of a right triangle! Do you know where I live?

Geometry Level pending

The circle x 2 + y 2 4 x 4 y + 4 = 0 x^2+y^2-4x-4y+4=0 is inscribed in a triangle which has two of its sides along the coordinate axes.The locus of the circumcentre of the triangle is

x + y x y + k x 2 + y 2 x+y-xy+k\sqrt{x^2+y^2}

The value of k k is?

1 3 2 4

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1 solution

Sanchayan Dutta
Sep 13, 2015

The equation of L is x / a + y / b = 1 x/a+y/b=1 and the distance of centre to the line is 2 / a + 2 / b 1 1 / a 2 + 1 / b 2 = 2 \frac{|2/a+2/b-1|}{\sqrt{1/a^2+1/b^2}}=2 .Now circumcentre is the point (a/2,b/2) since it bisects line L between the axes (since it forms a right angle triangle).Slightly manipulating the equation we get a / 2 + b / 2 ( a / 2 ) ( b / 2 ) ( a / 2 ) 2 + ( b / 2 ) 2 = 1 \frac{a/2+b/2-(a/2)(b/2)}{\sqrt{(a/2)^2+(b/2)^2}}=-1 Which matches with the equation in question.Hence k=1.Replace a/2 with x and b/2 with y to get that equation.

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