Question

Show that the points $A(1,,0),B(2,-7),C(8,1)andD(9,-6)$ all lie on the same circle. Find the equation of this circle, its centre and radius.

Answer 1

$A(1,0)B(2,-7),C(8,1)D(9,-6)$. Let Centre $(x,y)$

$\sqrt{(x-1{)}^2(y-0{)}^2}=\sqrt{(x-2{)}^2+(y+7{)}^2}=\sqrt{(x-8{)}^2+(y-1{)}^2}$

$(x-1{)}^2+y^2=(x-2)+(y+7{)}^2\Rightarrow x^2+1-2x+y^2+y^2=x^2+4-4x+y^2+49+14y-52+2x-14y$......(1)

$x^2+y-42+y^2+49+14y=x^2+6y-16x+y^2+1-2y$

$\Rightarrow -12+12x+16y=0$......(2)

Solving (1) and (2)

$\Rightarrow 12x-14y=52\times 6$

$12x+16y=12$

- - -

$\overline{-100y=300}\Rightarrow y=-3$

$-12+12x-48=0\Rightarrow 12x=60\Rightarrow x=5$

Centre $(5,-3)$

$radius=\sqrt{(5-1{)}^2+(-3{)}^2}=5$

eq. of circle $=(x-5{)}^2+(y+3{)}^2=25$

Now, ${}^{\prime }{D}^{\prime }$ should be this circle $(9.-6)$

$(9-5{)}^2+(-6+3{)}^2=4^2+3^2=25$