Problem 1 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 4, find an equation of the circle with center at \(C\) and radius \(r\). Write the equation in both the centerradius form and the general form. $$ C(4,-3), r=5 $$

Step-by-Step Solution

Answer

Center-radius form: \[ (x-4)^2 + (y+3)^2 = 25 \]. General form: \[ x^2 + y^2 - 8x + 6y = 0 \].

Step 1 - Write the center-radius form of the circle equation

The center-radius form of the circle equation is written as \( (x - h)^2 + (y - k)^2 = r^2 \), where \(C(h, k)\) is the center of the circle and \(r\) is the radius. For this problem, \(C(4, -3)\) and \(r = 5\). Thus, the equation becomes \[ (x - 4)^2 + (y + 3)^2 = 25 \].

Step 2 - Expand the center-radius form to general form

The general form of the circle equation is \(Ax^2 + Ay^2 + Dx + Ey + F = 0\). To convert the center-radius form \[ (x - 4)^2 + (y + 3)^2 = 25 \] into the general form, expand and simplify the equation: \lbrace (x - 4)^2 + (y + 3)^2 = 25 \: (x^2 - 8x + 16) + (y^2 + 6y + 9) = 25 \: x^2 + y^2 - 8x + 6y + 25 = 25 \: x^2 + y^2 - 8x + 6y = 0\rbrace.

Key Concepts

Center-Radius Form

The center-radius form of a circle's equation is one of the simplest ways to describe a circle. It highlights the key properties of the circle: its center and radius. This form is expressed as \[ (x - h)^2 + (y - k)^2 = r^2 \]here:

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

For example, if a circle has a center at \(C(4, -3)\) and a radius of \5\, we plug these values into the center-radius form equation to get: \[ (x - 4)^2 + (y + 3)^2 = 25 \] This form is useful because it directly tells us where the circle is located and how big it is.

General Form

The general form of a circle's equation is another way to express the equation of a circle. It looks different from the center-radius form and is written as: \[ Ax^2 + Ay^2 + Dx + Ey + F = 0 \] This form can sometimes make it easier to manipulate or compare equations but requires more steps to identify the center and radius. To convert the center-radius form equation \[ (x - 4)^2 + (y + 3)^2 = 25 \] into the general form, we need to expand and simplify the equation. Starting from \[ (x - 4)^2 + (y + 3)^2 = 25, \ (x^2 - 8x + 16) + (y^2 + 6y + 9) = 25 \] and combining like terms, we get: \[ x^2 + y^2 - 8x + 6y + 25 = 25, \ x^2 + y^2 - 8x + 6y = 0. \] This is the general form of the circle's equation.

Circle Equation Expansion

Expanding the circle’s equation involves converting the neat, compact center-radius form into the more spread out general form. The process of expansion requires applying algebraic operations to each term in the equation. Starting from: \[ (x - 4)^2 + (y + 3)^2 = 25 \]this involves expanding both squares: \[ (x - 4)^2 = x^2 - 8x + 16 \]and \[ (y + 3)^2 = y^2 + 6y + 9 \]. Combining and simplifying these, we have: \[ x^2 - 8x + 16 + y^2 + 6y + 9 = 25 \]. Finally, by moving all terms to one side of the equation and setting it to zero: \[ x^2 + y^2 - 8x + 6y = 0. \] This series of steps lets you transform an equation between forms, making it easier for various mathematical purposes.