The question asks which method is used to derive the equation of a circle in standard form.

Let's break down the options:

A. Quadratic formula - This formula is used to solve quadratic equations, not to derive the equation of a circle. So, this is incorrect.

B. Completing the square - This method is commonly used to derive the equation of a circle. The general form of a circle's equation is $x^2 + y^2 + Dx + Ey + F = 0$, and completing the square on the $x$- and $y$-terms transforms this into the standard form $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius. So, this is correct.

C. Distance formula - The distance formula is used to calculate the distance between two points, but it's not directly used to derive the equation of a circle. This is incorrect.

D. Midpoint formula - The midpoint formula is used to find the midpoint between two points, not to derive the equation of a circle. This is incorrect.

Correct answer: B. Completing the square is the method used to derive the equation of a circle in standard form.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you complete the square to derive the equation of a circle?
2. What is the general form of a circle's equation before completing the square?
3. How is the center and radius of a circle identified from its equation?
4. Can the distance formula be used to derive the radius of a circle given two points?
5. How does the standard form of a circle’s equation differ from its general form?

Tip: When working with equations of circles, always check if you need to complete the square to convert from general form to standard form!