Answer: (1/3,2)
Step-by-step explanation:
Let O(x,y) be the circumference of the triangle PQR,
Thus, by the property of circumcenter all the vertices of the triangle are at same distance from this circumcenter.
Thus, PO = QO
By the distance formula,
[tex](x+2)^2+(y-5)^2 = (x-4)^2+(y-1)^2[/tex]
[tex]\impliesx^2+ 4x + 4 + y^2+ 25 - 10 y = x^2+16 - 8 x + y^2+1 - 2y[/tex]
[tex]\implies 4x + 4 + 25 - 10 y = 16 - 8 x + 1 - 2y[/tex]
[tex]\implies 4x +8x-10y+2y = 16 + 1 -25-4[/tex]
[tex]\implies 12x - 8y = 17 - 29[/tex]
[tex]\implies 12 x - 8 y = - 12[/tex]---------(1)
Similarly, QO = RO
[tex]\implies (x-4)^2 + (y-1)^2 = (x+2)^2+(y+3)^2[/tex]
[tex]\implies x^2 + 16 - 8x + y^2+ 1 - 2y = x^2+ 4 + 4 x +y^2+ 9 + 6 y[/tex]
[tex]\implies 16 - 8x + 1 - 2y = 4 + 4 x + 9 + 6 y[/tex]
[tex]\implies - 8x -4x-2y-6y=4+9-16-1[/tex]
[tex]\implies -12 x - 8 y = 13 - 17[/tex]
[tex]\implies 12 x + 8 y = 4[/tex] ------------(2)
By adding equation (1) and (2),
24 x = - 8
x = 1/3
By putting this value in equation (1),
We get,
Thus, 4 - 8 y = -12
8 y = 16
y = 2
