The locus of the incentre of the triangle formed by the lines x y-4 x-4 y+16=0 and x+y=5 is

Step-by-step solution

x y-4 x-4 y+16=0 aligned & x(y-4)-4(y-4)=0 & (y-4)(x-4)=0 aligned The 2 lines are x=4 and y=4 aligned & L ₁: x=4 & ~L ₂: y=4 & ~L ₃: x+y=5 aligned aligned & AB = c =3 & BC = a =3 2 & CA = b =3 aligned aligned & A (x₁ y₁ ) (4,4) & B (x₂, y₂ ) (4,1) & C (x₃, y₃ ) (1,4) aligned Incentre = ( a x₁+b x₂+c x₃ a+b+c , a y₁+b y₂+c y₃ a+b+c ) aligned & = 3 2 4+3 4+3 1 3+3+3 2 , 3 2 4+3.1+3 4 3 2 +3+3 & = 15+12 2 6+3 2 , 15+12 2 6+3 2 & = 5+4 2 2+ 2 , 5+4 2 2+ 2 aligned Co-ordinates of x and y satisfy x=y locus of incentre is