The distance from the origin to the orthocentre of the triangle formed by the lines x+y-1=0 and 6 x^2-13 x y+5 y^2=0 is

Step-by-step solution

Given lines are x+y-1=0 and 6 x^2-13 x y+5 y^2=0 array lc & 6 x^2-10 x y-3 x y+5 y^2=0 & 2 x(3 x-5 y)-y(3 x-5 y)=0 & (2 x-y)(3 x-5 y)=0 & 2 x-y=0 or & 3 x-5 y=0 array Let orthocentre be (h, k) . Slope of O P slope of A B=-1 k h -1=-1 Now, slope of O B slope of A D=-1 array ll & 2 ( 3 8 -k 5 8 -h )=-1 3 4 -2 h=h- 5 8 & 3 4 + 5 8 =3 h 6+5 8 =3 h & 11 8 =3 h h= 11 24 & k= 11 24 array Now, O P= h^2+k^2 = ( 11 24 )^2+ ( 11 24 )^2 = 11 24 2