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 quadratic formula
b. the equation of a circle can be derived by solving a quadratic equation using the method of completing the square
c. the equation of a circle can be derived using the midpoint formula
d. the equation of a circle can be derived using the distance formula

Explanation:

To determine the correct statement for deriving the equation of a circle in standard form:

• Option A: The quadratic formula is used to solve quadratic equations, not to derive the circle's equation. Eliminate A.
• Option B: Completing the square is used to convert the general form of a circle's equation to standard form, not the primary method of derivation. Eliminate B.
• Option C: The midpoint formula is for finding the midpoint of a segment, not related to deriving the circle's equation. Eliminate C.
• Option D: The equation of a circle is derived from the distance formula. A circle is the set of all points \((x,y)\) equidistant (radius \(r\)) from the center \((h,k)\). The distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) gives \(r=\sqrt{(x - h)^2+(y - k)^2}\), squaring both sides gives the standard form \((x - h)^2+(y - k)^2=r^2\). So D is correct.

Answer:

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