Exploring Coordinates of Circle & the Standard Equation

Circles have been previously studied in this course. In this lesson, the equation of a circle on a coordinate plane will be derived. Furthermore, when given the equation of a circle, its center and radius will be found, and its graph will be drawn.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Circles
Distance Formula
Completing the square
Solving equations

Try your knowledge on these topics.

a By using the Distance Formula, find the distance between points

P

and

Q .

b Identify the center and find the radius of the circle below.

c By completing the square, determine the values of

h

and

k .

x^{2} + 6 x + 11 = (x - h)^{2} + k

d Solve the linear equation.

3 x + 8 = 53

e Solve the quadratic equation.

x^{2} + 5 x = 50

Write the smaller solution first.

Challenge

Investigating Points on a Circle's Circumference

In the diagram below, a circle centered at the origin with radius

5

has been drawn. Some points that lie on the circle can be identified.

Given that

P

is a point that lies on the circle and the

x -

coordinate of

P

1

, can you determine the

y -

coordinate of

P ?

Discussion

Deriving the Equation of a Circle at the Origin

The equation of a circle on a coordinate plane can be obtained using the Distance Formula. For example, consider a circle centered at the origin with a radius of

2 .

An arbitrary point

P (x, y)

lies on the circle.

Since the radius is

2,

the distance between

O (0, 0)

and

P (x, y)

is also

2 .

This information can be substituted into the Distance Formula.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

2 = (x - 0)^{2} + (y - 0)^{2}

Simplify

SubTerms

Subtract terms

2 = x^{2} + y^{2}

RaiseEqn

$LHS^{2} = RHS^{2}$

4 = x^{2} + y^{2}

RearrangeEqn

Rearrange equation

x^{2} + y^{2} = 4

The equation of the given circle was obtained.

$x^{2} + y^{2} = 4$

Explore

Standard Equation of a Circle

The result previously obtained can be generalized to find the equation of a circle with a certain center and given radius.

On a coordinate plane, consider a circle with radius $r$ and center $(h, k) .$

The standard equation of the above circle is given below.

(x - h)^{2} + (y - k)^{2} = r^{2}

Proof

On the diagram below, consider an arbitrary point on a circle with coordinates

(x, y) .

As it can be seen in the diagram, the distance between the center

(h, k)

and the point

(x, y)

r .

This information can be substituted into the Distance Formula.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

SubstituteValues

Substitute values

r = (x - h)^{2} + (y - k)^{2}

Simplify

RaiseEqn

$LHS^{2} = RHS^{2}$

r^{2} = (x - h)^{2} + (y - k)^{2}

RearrangeEqn

Rearrange equation

(x - h)^{2} + (y - k)^{2} = r^{2}

The equation of a circle with radius

r

and center

(h, k)

was obtained by using the Distance Formula.

By using this formula, the standard equation of any circle drawn on a coordinate plane can be written.

Example

Finding the Standard Equation of a Circle

Tearrik has one last problem to solve before going to a BBQ. He needs to find the standard equation of the circle shown below.

Tearrik remembers that the standard equation of a circle is $(x - h)^{2} + (y - k)^{2} = r^{2} .$ However, he does not remember how to find the values of $h,$ $k,$ and $r .$ Help Tearrik get to the BBQ by finding these values!

Hint

The center of the circle is $(h, k)$ and its radius $r .$

Solution

In the standard equation of a circle,

(h, k)

is the center of the circle. Therefore, by noting the coordinates of the center in the diagram, the values of

h

and

k

can be determined. Furthermore, the value of

r

is given by the distance between the center and any point on the circle.

The center of the circle can be seen to have the coordinates

(- 1, 2) .

Therefore, it can be stated that

h = - 1

and

k = 2 .

It can also be seen that the radius of the circle is

r = 5 .

By substituting these values into the standard equation of a circle, the equation of the given circle can be obtained.

(x - h)^{2} + (y - k)^{2} = r^{2}

SubstituteValues

Substitute values

(x - (- 1))^{2} + (y - 2)^{2} = 5^{2}

Simplify

SubNeg

$a - (- b) = a + b$

(x + 1)^{2} + (y - 2)^{2} = 5^{2}

CalcPow

Calculate power

(x + 1)^{2} + (y - 2)^{2} = 25

Example

Finding the Center and Radius of a Circle

Just like her classmate, Zain has one last problem to solve before getting to go to the BBQ. She has been asked to identify the center and the radius of the circle whose standard equation is given below.

(x - 4)^{2} + (y + 3)^{2} = 25

Zain has also been asked to graph the circle on a coordinate plane. Help Zain get to the BBQ!

Hint

The standard equation of a circle is $(x - h)^{2} + (y - k)^{2} = r^{2} .$ Here, the center of the circle is $(h, k)$ and its radius is $r .$

Solution

The standard equation of a circle is shown below.

(x - h)^{2} + (y - k)^{2} = r^{2}

Here, the center of the circle is

(h, k)

and its radius is

r .

Therefore, it is convenient to rewrite the given equation to match this format.

(x - 4)^{2} + (y + 3)^{2} = 25

Rewrite

3

- (- 3)

and Rewrite

25

5^{2}

$a + b = a - (- b)$

(x - 4)^{2} + (y - (- 3))^{2} = 25

WritePow

Write as a power

(x - 4)^{2} + (y - (- 3))^{2} = 5^{2}

From the obtained equation, the center of the circle can be identified as

(4, - 3)

and its radius as

5 .

With this information, the circle can be drawn on a coordinate plane.

Example

Rewriting the Equation of a Circle

Sometimes the equation of a circle needs a significant change to be rewritten as the standard equation of a circle. Typically in those cases, the equation can be rewritten by completing the square.

To be allowed to help design her schools basketball court, Dominika was asked to identify the center and the radius of the circle whose equation is given below.

x^{2} - 2 x + y^{2} = 4

By rewriting the above equation as the standard equation of a circle, identify the center and the radius. If any answer is an irrational number, write its exact value.

Hint

Add a number to the expression $x^{2} - 2 x$ so that it becomes a perfect square trinomial.

Solution

To rewrite the given equation as the standard equation of a circle, a method called completing the square can be used. This method consists in writing an expression as a perfect square trinomial so that it can be factored as the square of a binomial. First, the expression

x^{2} - 2 x

will be considered.

x^{2} - 2 x + y^{2} = 4

Ignoring the negative sign, the coefficient of the variable

x

2 .

This coefficient is commonly called

b .

x^{2} - 2 x + y^{2} = 4

Next, the term

(\frac{b}{2})^{2}

will be added to and subtracted from the expression. Note that adding and subtracting the same value does not modify the expression. Since

b = 2,

the value of

(\frac{2}{2})^{2}

can be calculated.

(\frac{2}{2})^{2}

Simplify

QuotOne

$\frac{a}{a} = 1$

1^{2}

BaseOne

$1^{a} = 1$

1

Therefore,

1

will be added to and subtracted from

x^{2} - 2 x .

Then, the resulting perfect square trinomial will be factored and written as the square of a binomial.

x^{2} - 2 x + y^{2} = 4

IdPropAdd

Identity Property of Addition

x^{2} - 2 x + 0 + y^{2} = 4

Rewrite

Rewrite $0$ as $1 - 1$

x^{2} - 2 x + 1 - 1 + y^{2} = 4

a^{2} - 2 a b + b^{2} = (a - b)^{2}

IdPropMult

Identity Property of Multiplication

x^{2} - 2 x (1) + 1 - 1 + y^{2} = 4

WritePow

Write as a power

x^{2} - 2 x (1) + 1^{2} - 1 + y^{2} = 4

FacNegPerfectSquare

$a^{2} - 2 a b + b^{2} = (a - b)^{2}$

(x - 1)^{2} - 1 + y^{2} = 4

The process of completing the square is now finished. Finally, to obtain the standard equation of the circle, the number

1

will be added to both sides of the equation and

y

will be written as

y - 0 .

Also, the resulting number on the right-hand side will be expressed as a square.

(x - 1)^{2} - 1 + y^{2} = 4

Simplify

AddEqn

$LHS + 1 = RHS + 1$

(x - 1)^{2} + y^{2} = 5

Rewrite

Rewrite $y$ as $y - 0$

(x - 1)^{2} + (y - 0)^{2} = 5

Rewrite

Rewrite $5$ as $(5)^{2}$

(x - 1)^{2} + (y - 0)^{2} = (5)^{2}

The standard equation of the circle was obtained. The center can be identified as

(1, 0)

and the radius as

5 .

Explore

Rewriting the Equation of One More Circle

This time, Dominika wants to play basketball on Sunday. Her father will be okay with that only if she completes her math homework. To do so, Dominika has to identify the center and the radius of the circle whose equation is given below.

x^{2} + 6 x + y^{2} - 4 y = - 3

By rewriting the above equation as the standard equation of a circle, identify the center and the radius. If any of the answers is an irrational number, write its exact value.

Hint

Complete the square for the $x -$ and the $y -$ variable.

Solution

To rewrite the given equation as the standard equation of a circle, the process of completing the square needs to be performed for both variables.

x^{2} + 6 x + y^{2} - 4 y = - 3

IdPropAdd

Identity Property of Addition

x^{2} + 6 x + 0 + y^{2} - 4 y + 0 = - 3

Rewrite

Rewrite $0$ as $9 - 9 & 4 - 4$

x^{2} + 6 x + 9 - 9 + y^{2} - 4 y + 4 - 4 = - 3

a^{2} \pm 2 a b + b^{2} = (a \pm b)^{2}

SplitIntoFactors

Split into factors

x^{2} + 2 (3) x + 9 - 9 + y^{2} - 2 (2) y + 4 - 4 = - 3

CommutativePropMult

Commutative Property of Multiplication

x^{2} + 2 x (3) + 9 - 9 + y^{2} - 2 y (2) + 4 - 4 = - 3

WritePow

Write as a power

x^{2} + 2 x (3) + 3^{2} - 9 + y^{2} - 2 y (2) + 2^{2} - 4 = - 3

$a^{2} \pm 2 a b + b^{2} = (a \pm b)^{2}$

(x + 3)^{2} - 9 + (y - 2)^{2} - 4 = - 3

Simplify

SubTerms

Subtract terms

(x + 3)^{2} + (y - 2)^{2} - 13 = - 3

AddEqn

$LHS + 13 = RHS + 13$

(x + 3)^{2} + (y - 2)^{2} = 10

$a + b = a - (- b)$

(x - (- 3))^{2} + (y - 2)^{2} = 10

Rewrite

Rewrite $10$ as $(10)^{2}$

(x - (- 3))^{2} + (y - 2)^{2} = (10)^{2}

The standard equation of the circle was obtained. The center can be identified as

(- 3, 2)

and the radius as

10 .

Closure

Determining a Point on the Circumference of a Circle

The challenge presented at the beginning of this lesson can be solved by writing the equation of the circle.

On a coordinate plane, a circle centered at the origin with radius $5$ was drawn. Also, a point on the circle with $x -$ coordinate $1$ was plotted.

By writing the standard equation of the circle, find the

y -

coordinate of

P .

Write the answer as an exact value.

Hint

The standard equation of a circle is $(x - h)^{2} + (y - k)^{2} = r^{2},$ where $(h, k)$ is the center and $r$ the radius.

Solution

Recall the standard equation of a circle.

(x - h)^{2} + (y - k)^{2} = r^{2}

Here,

(h, k)

is the center and

r

the radius of the circle. Knowing that the center of the circle drawn in the diagram is

(0, 0)

and that its radius is

5,

its standard equation can be written.

(x - h)^{2} + (y - k)^{2} = r^{2}

SubstituteValues

Substitute values

(x - 0)^{2} + (y - 0)^{2} = 5^{2}

Simplify

SubTerms

Subtract terms

x^{2} + y^{2} = 5^{2}

CalcPow

Calculate power

x^{2} + y^{2} = 25

Recall that the

x -

coordinate of

P

1 .

Therefore, to find its

y -

coordinate, this value can be substituted into the equation of the circle.

x^{2} + y^{2} = 25

Substitute

$x = 1$

1^{2} + y^{2} = 25

Solve for

y

BaseOne

$1^{a} = 1$

1 + y^{2} = 25

SubEqn

$LHS - 1 = RHS - 1$

y^{2} = 24

SqrtEqn

$LHS = RHS$

y = \pm 24

SplitIntoFactors

Split into factors

y = \pm 4 (6)

SqrtProd

$a \cdot b = a \cdot b$

y = \pm 46

CalcRoot

Calculate root

y = \pm 26

The

y -

coordinate of

P

can be either

26

- 26 .

However, from the diagram it can be observed that

P

is located in Quadrant I, where all values of the

y -

variable are positive. Therefore, the

y -

coordinate of

P

26 .

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Catch-Up and Review

Challenge

Investigating Points on a Circle's Circumference

Discussion

Deriving the Equation of a Circle at the Origin

Explore

Standard Equation of a Circle

Proof

Example

Finding the Standard Equation of a Circle

Hint

Solution

Example

Finding the Center and Radius of a Circle

Hint

Solution

Example

Rewriting the Equation of a Circle

Hint

Solution

Explore

Rewriting the Equation of One More Circle

Hint

Solution

Closure

Determining a Point on the Circumference of a Circle

Hint

Solution

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