Question

Multiple Choice The equation of a circle can be changed from general form to standard from by doing which of the following? (a) completing the squares (b) solving for \(x\) (c) solving for \(y\) (d) squaring both sides

Short answer

Completing the squares.

Step-by-step solution

Identify the General Form of a Circle

The general form of a circle's equation is given by: Ax^2 + Ay^2 + Dx + Ey + F = 0 where A, D, E, and F are constants.

Recognize the Standard Form of a Circle

The standard form of a circle's equation is: (x-h)^2 + (y-k)^2 = r^2 where (h, k) is the center and r is the radius.

Notice the Transformation

To transform the general form to the standard form, recognize that completing the squares is used. This method rearranges and groups terms involving x and y to form perfect square trinomials.

Check the Options

Among the options given: (a) completing the squares (b) solving for x (c) solving for y (d) squaring both sides, option (a) completing the squares is the correct choice.

Key concepts

General Form

The general form of a circle's equation is a fundamental concept in geometry. It is presented as: \[Ax^2 + Ay^2 + Dx + Ey + F = 0\] where *A*, *D*, *E*, and *F* are constants. This form can be a bit confusing at first because it doesn't immediately reveal the circle's center or radius. However, it consolidates the information needed to describe any circle in a linear equation format. You can think of these constants as parameters that mold the circle into different shapes and locations within the coordinate plane.

Standard Form

The standard form of a circle's equation is more intuitive once you are familiar with the format: \[(x-h)^2 + (y-k)^2 = r^2\] In this form,

• *(h, k)* represents the center of the circle.
• *r* is the radius.

This format straightforwardly shows where the circle is positioned and how big it is. Unlike the general form, the standard form makes it easy to identify these characteristics. Transforming the general form into the standard form often involves rearranging and simplifying terms, which brings us to the next crucial technique.

Completing the Square

Transforming a circle's equation from the general form to the standard form involves a method known as completing the square. This technique reshapes parts of the equation into perfect square trinomials. Here's a simplified outline of how you can do that:

• Start with the general form: \[Ax^2 + Ay^2 + Dx + Ey + F = 0\]
• Group the terms involving *x* and *y*: \[Ax^2 + Dx + Ay^2 + Ey = -F\]
• To complete the square, you'll need to factor out *A* from x-terms and y-terms, if *A* isn't 1. Then re-arrange to form perfect squares.

This step transforms your equation into something manageable and in the standard form. By mastering this technique, you'll be able to tackle various problems involving circles with confidence.