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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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The Equation of a Circle

A circle is the set of all points equidistant from a single fixed point in a plane. The single fixed point is called the center of the circle. The distance from the center to any point on the circle is called the radius.

This circle is drawn on a Cartesian coordinate system.

In the figure:

• The ordered pair (h, k) is the center of the circle.

• The ordered pair (x, y) represents any point on the circle.

• The radius, r, is the distance between the center of the circle, (h, k), and any point on the circle, (x, y).

Now we may use the distance formula to express the distance between the center of the circle and any point on the circle. That distance is the radius of the circle.
In the distance formula, make the following substitutions:
distance = r

(x1, y1) = (h, k)

(x2, y2) = (x, y)

To remove the square root, square both sides.  r2 = (x - h)2 + (y - k)2

The result is the equation of a circle.

This equation is usually written with r2 on the right.

Formula — The Equation of a Circle

The equation of a circle in the xy-plane is given by

(x - h)2 + (y - k)2 = r2

where the point (h, k) is the center of the circle, (x, y) is any point on the circle, and r is the radius of the circle.

Example 1

Find the equation of the circle with center at (-4, 3) and radius 2.

Solution (x - h)2 + (y - k)2 = r2
In the equation of a circle, substitute -4 for h, 3 for k, and 2 for r.

Simplify.

(x - (-4))2 + (y - 3)2 = 22

(x + 4)2 + (y - 3)2 = 4

The equation of the circle is (x + 4)2 + (y - 3)2 = 4.

Example 2

Find the center and the radius of the circle whose equation is (x - 5)2 + (y + 3)2 = 16.

Solution

The given equation has almost the same

form as the standard equation of a circle.

x - 5 has the form x - h. Therefore, h is 5.

y + 3 does not have the form y - k. Rewrite y + 3 as y - (-3).

Therefore, k is -3.

r2 is 16.

Therefore, r is

(x - h)2 + (y - k)2 = r2

(x - 5)2 + (y + 3)2 = 16

 

(x - 5)2 + (y - (-3))2 = 16

The center of the circle, (h, k), is (5, -3). The radius, r, is 4.

Example 3

Find the center and the radius of the circle with equation (x + 1)2 + y2 = 20.

Solution

The given equation has almost the same

form as the standard equation of a circle.

Rewrite x - 1 as x - (-1).

Rewrite y2 as (y - 0)2.

Now, we can identify h, k, and r.

(x - h)2 + (y - k)2 = r2

(x + 1)2 + y2 = 20

[x - (-1)]2 + y2 = 20

[x - (-1)]2 + (y - 0)2 = 20

x - (-1) has the form x - h. Therefore, h is -1.

y - 0 has the form y - k. Therefore, k is 0.

r2 is 20. Therefore, r is

The center of the circle, (h, k), is (-1, 0). The radius is

Note:

We can also write