If 2 x^2-3 x y+y^2=0 represents two sides of a triangle and x+y-1=0 is its third side, then the distance between the orthocenter and the circumcentre of that triangle is

Step-by-step solution

Given pair of equations aligned & 2 x^2-3 x y+y^2=0 & (x-y)(2 x-y)=0 & x-y=0 and 2 x-y=0 aligned aligned & O A= 1 4 + 1 4 = 1 2 & O B= 1 9 + 4 9 = 5 3 & A B= ( 1 3 - 1 2 )^2+ ( 2 3 - 1 2 )^2 = 2 6 & O A^2+A B^2= 1 2 + 2 36 = 5 9 = ( 5 3 )^2=O B^2 aligned O A B is right angle triangle, where O B is hypotenuse. Distance between orthocentre and circumcentre = Hypotenuse 2 = 5 6 .