Question

If the points $A(4,3)$ and $B(x,5)$ are on the circle with the centre $O(2,3)$. Find the value of $x$.

Answer 1

Let equation of circle be

$x^2+y^2+2gx+2fy+c=0$

$(-g,-f)\to$ centre is $(2,3)$

$x^2+y^2+2(-2)\left(x\right)+2(-3)y+c=0$

$A(4,3)$ lies on circle.

$(4{)}^2+(3{)}^3-4\left(4\right)-6\left(3\right)+c=0$

$c=9$

$\therefore x^2+y^2-4x-6y+9=0$ is equation of circle.

$B(x,5)$

$x^2+25-4x-6\left(5\right)+9=0$

$x^2-4x+4=0$

$(x-2{)}^2=0\Rightarrow x=2$