Question

which statement describes how to derive the equation of a circle in standard form? choose the correct statement a. the equation of a circle can be derived using the distance formula b. the equation of a circle can be derived using the quadratic formula c. the equation of a circle can be derived by solving a quadratic equation using the method of completing the square d. the equation of a circle can be derived using the midpoint formula

Explanation:

The standard - form equation of a circle \((x - h)^2+(y - k)^2=r^2\) is derived from the distance formula. The distance between a point \((x,y)\) on the circle and the center \((h,k)\) of the circle is the radius \(r\). By the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), substituting \((x_1,y_1)=(h,k)\), \((x_2,y_2)=(x,y)\) and \(d = r\), we get \(\sqrt{(x - h)^2+(y - k)^2}=r\), and then squaring both sides gives the standard - form equation of the circle. The quadratic formula is used to solve quadratic equations \(ax^{2}+bx + c = 0\), completing the square is also for solving or rewriting quadratic equations, and the mid - point formula is for finding the mid - point between two points, none of which are used for deriving the circle's standard form equation.

Answer:

A. The equation of a circle can be derived using the distance formula