Problem 4 - 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. $$ C(-1,1), r=2 $$

Step-by-Step Solution

Answer

Center-radius form: \ (x + 1)^2 + (y - 1)^2 = 4 \; General form: \ x^2 + y^2 + 2x - 2y - 2 = 0 \

Step 1: Identify the standard form

The standard form for the equation of a circle with center \(C(h,k)\) and radius \(r\) is \[ (x - h)^2 + (y - k)^2 = r^2 \]

Step 2: Plug in the given center and radius

Given center \(C(-1,1)\) and radius \(r=2\), substitute \(h=-1\), \(k=1\), and \(r=2\) into the standard form equation: \[ (x + 1)^2 + (y - 1)^2 = 2^2 \]

Step 3: Simplify the radius term

Calculate the square of the radius: \(2^2 = 4\). The equation becomes: \[ (x + 1)^2 + (y - 1)^2 = 4 \]

Step 4: Expand to general form

Expand \( (x + 1)^2 \) and \( (y - 1)^2 \) and combine like terms to convert to general form: \[ (x^2 + 2x + 1) + (y^2 - 2y + 1) = 4 \] which simplifies to: \[ x^2 + y^2 + 2x - 2y + 2 = 4 \]

Step 5: Further simplify the general form

Subtract 4 from both sides to get the general form: \[ x^2 + y^2 + 2x - 2y - 2 = 0 \]

Key Concepts

Standard Form of a Circle

In coordinate geometry, the standard form of the equation of a circle is key to understanding its properties. This form is derived from the circle's center and radius. The standard form is expressed as:

o[x - h)^2 + (y - k)^2 = r^2o

oHere, (h, k) represents the coordinates of the circle's center, and r denotes the radius. When given specific values, like a center at C(-1,1) and radius r=2, you plug these into the standard form to get: (x + 1)^2 + (y - 1)^2 = 4

This approach clearly shows the circle's location and size in the coordinate plane. It is ideal for quickly identifying the circle's basic properties.

General Form of a Circle

The general form of a circle's equation is often used for algebraic manipulation. This form is given by:

o[x^2 + y^2 + Dx + Ey + F = 0o

where D, E, and F are constants that result from expanding and simplifying the standard form. To transform a standard equation into its general form, follow these steps:

oExpand both squared terms in the standard formo
For the equation (x + 1)^2 + (y - 1)^2 = 4: expand to get x^2 + 2x + 1 + y^2 - 2y + 1. Simplify and combine like terms to get the general form: x^2 + y^2 + 2x - 2y + 2 = 4.
Subtract 4 from both sides to keep the right side as zero: x^2 + y^2 + 2x - 2y - 2 = 0.

This standard process ensures an accurate and useful equation for more complex algebraic operations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric shapes using a coordinate system. Circles in coordinate geometry are defined by their center and radius. When solving problems involving circles, remember these points:

• The equation of a circle helps to identify its position and size.
The standard form (x - h)^2 + (y - k)^2 = r^2 quickly reveals the center and radius.
The conversion to general form makes it easier to integrate with other algebraic expressions.

Understanding these forms and their applications in solving geometry problems is crucial for mastering the topic. It paves the way for tackling diverse and advanced questions in coordinate geometry.