Question

How is the distance formula related to the equation of a circle?

Answer 1

${(x-h)^{2}+(y-k)^{2}}= r^{2}$ is the required equation of a circle .

Step-by-step explanation:

We know that distance formula is the distance between any two points $p(x_1, y_1) , q(x_2,y_2)$ is given by:

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

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

⇒ $Distance^{2} = {(x_2-x_1)^{2}+(y_2-y_1)^{2}}$

Now, In equation of circle , Suppose we have center at $c(h,k)$  and arbitrary point in the circle is $r(x,y)$  . Distance between any point at circle to it's center is radius of circle i.e. r  , putting these values in distance formula we get:

⇒ $Distance^{2} = {(x_2-x_1)^{2}+(y_2-y_1)^{2}}$

⇒ $r^{2} = {(x-h)^{2}+(y-k)^{2}}$

⇒ ${(x-h)^{2}+(y-k)^{2}}= r^{2}$ , which the required equation of a circle .