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

Use the distance formula

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

That means, $AO=BO$ ($\because$ all the radii of a circle are equal )

According to the distance formula, the distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by:

$AB=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Let's use the distance formula to substitute the given coordinates in $AO=BO$:

$\Rightarrow$$\sqrt{{(2-4)}^2+{(3-3)}^2}$ $=\sqrt{{(2-x)}^2+{(3-5)}^2}$

$\Rightarrow$ ${(2-4)}^2+{(3-3)}^2$ $={(2-x)}^2+{(3-5)}^2$ [squaring both sides]

$\Rightarrow$ $4$ $={\left(2-x\right)}^2+4$

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

$\Rightarrow$ $x=2$

Hence, the value of $x$ is $2$.