Question

The coordinates of the centre of a circle passing through $(1,2),(3,-4)and(5,-6)$ is

• A. $(2,11)$
• B. $(11,2)$
• C. $(11,-2)$
• D. $(-2,11)$

Answer 1

The correct option is B $(11,2)$

Given three points $(1,2),(3,-4)and(5,-6)$

Let the coordinates o centre be $O(h,k)$

The distance from the centre to a point on the circumference is equal to radius.

$\therefore$ $OA=OB=OC$

From distance formula

$O{A}^2=O{B}^2(h-1{)}^2+(k-2{)}^2=h-3{)}^2+(k+4{)}^2{h}^2-2h+1+k^2-4k+4=h^2-6h+9+k^2+8k+16-2h-4k+5=-6h+8k+254h-12k=20⟹h-3k=5⟶(1)O{B}^2=O{C}^2(h-3{)}^2+(k+4{)}^2=(h-5{)}^2+(k+6{)}^2{h}^2-6h+9+k^2+8k+16=h^2-10h+25+k^2+12k+36-6h+8k+25=-10h+12k+61⟹4h-4k=36⟹h-k=9⟶(2)\frac{h-3k=5-h\pm k=-9}{-2k=-4}\therefore k=2\therefore h=11$

$\therefore$ The Coordinates are $(h,k)=(11,2)$