Question

how can the equation of a circle be derived?
1 of 4 questi
set the circles center equal to any point on the circle and simplify.
set the distance from the circles center to the axis of symmetry equal to the radius and simplify.
set the distance from the circles center to any point o the circle equal to the radius and simplify.
use the midpoint formula and the circles center to relate any two points on opposite sides of the circle.

Explanation:

Step1: Recall the definition of a circle

A circle is the set of all points \((x,y)\) in a plane that are at a given distance \(r\) (the radius) from a given point \((h,k)\) (the center).

Step2: Use the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let \((x_1,y_1)=(h,k)\) (center) and \((x_2,y_2)=(x,y)\) (a point on the circle). Since the distance \(d = r\), we have \(r=\sqrt{(x - h)^2+(y - k)^2}\).

Step3: Square both sides to simplify

Squaring both sides of the equation \(r=\sqrt{(x - h)^2+(y - k)^2}\) gives \(r^{2}=(x - h)^2+(y - k)^2\), which is the standard equation of a circle. This process is equivalent to setting the distance from the circle's center \((h,k)\) to any point \((x,y)\) on the circle equal to the radius \(r\) and simplifying.

Answer:

Set the distance from the circle's center to any point on the circle equal to the radius and simplify.