Answer:
r^2 = (x - h)^2 + (y - k)^2
Explanation:
To derive the equation of a circle using the distance formula, we start with the distance formula itself. The distance formula gives us the distance between two points (x1, y1) and (x2, y2) in a two-dimensional plane, and it is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Now, let's consider a circle with center (h, k) and a point on the circle (x, y). We want to find the equation that relates these points.
We know that the distance between the center (h, k) and any point (x, y) on the circle is equal to the radius (r) of the circle. So, we can use the distance formula to express this relationship:
r = √((x - h)^2 + (y - k)^2)
Here, (x, y) represents any point on the circle, and (h, k) represents the center of the circle.
To solve for the general equation of a circle, we can square both sides of the equation to eliminate the square root:
r^2 = (x - h)^2 + (y - k)^2
This equation, r^2 = (x - h)^2 + (y - k)^2, is the general equation of a circle with center (h, k) and radius r.
It represents all the points (x, y) that are equidistant from the center (h, k) with a distance equal to the radius (r).
Hope this helps!