Circles

Mathematics 4

August 10, 2011

Mathematics 4 () Circles August 10, 2011 1 / 17

Review of Completing the Square

Completing the Square

Express the following quadratic expressions as (y − k) = a(x− h)2.

1 x2 − y − 12x+ 7 = 0

→ (y + 29) = (x− 6)2

2 2x2 − 5x− y − 3 = 0

→ y + 498 = 2(x− 5

4)2

Mathematics 4 () Circles August 10, 2011 2 / 17

Review of Completing the Square

Completing the Square

Express the following quadratic expressions as (y − k) = a(x− h)2.

1 x2 − y − 12x+ 7 = 0

→ (y + 29) = (x− 6)2

2 2x2 − 5x− y − 3 = 0

→ y + 498 = 2(x− 5

4)2

Mathematics 4 () Circles August 10, 2011 2 / 17

Review of Completing the Square

Completing the Square

Express the following quadratic expressions as (y − k) = a(x− h)2.

1 x2 − y − 12x+ 7 = 0

→ (y + 29) = (x− 6)2

2 2x2 − 5x− y − 3 = 0

→ y + 498 = 2(x− 5

4)2

Mathematics 4 () Circles August 10, 2011 2 / 17

Review of Completing the Square

Completing the Square

Express the following quadratic expressions as (y − k) = a(x− h)2.

1 x2 − y − 12x+ 7 = 0

→ (y + 29) = (x− 6)2

2 2x2 − 5x− y − 3 = 0

→ y + 498 = 2(x− 5

4)2

Mathematics 4 () Circles August 10, 2011 2 / 17

Review of Completing the Square

Completing the Square

Express the following quadratic expressions as (y − k) = a(x− h)2.

1 x2 − y − 12x+ 7 = 0

→ (y + 29) = (x− 6)2

2 2x2 − 5x− y − 3 = 0

→ y + 498 = 2(x− 5

4)2

Mathematics 4 () Circles August 10, 2011 2 / 17

Review of Completing the Square

Completing the Square

Express the following expressions as (x− h)2 + (y − k)2 = r, where h, k,and r are constants.

1 x2 + y2 + 2x− 8y + 4 = 0

→ (x+ 1)2 + (y − 4)2 = 13

2 9x2 + 9y2 + 6x− 12y + 5 = 63

→ (x+ 13)

2 + (y − 23)

2 = 7

Mathematics 4 () Circles August 10, 2011 3 / 17

Review of Completing the Square

Completing the Square

Express the following expressions as (x− h)2 + (y − k)2 = r, where h, k,and r are constants.

1 x2 + y2 + 2x− 8y + 4 = 0

→ (x+ 1)2 + (y − 4)2 = 13

2 9x2 + 9y2 + 6x− 12y + 5 = 63

→ (x+ 13)

2 + (y − 23)

2 = 7

Mathematics 4 () Circles August 10, 2011 3 / 17

Review of Completing the Square

Completing the Square

Express the following expressions as (x− h)2 + (y − k)2 = r, where h, k,and r are constants.

1 x2 + y2 + 2x− 8y + 4 = 0

→ (x+ 1)2 + (y − 4)2 = 13

2 9x2 + 9y2 + 6x− 12y + 5 = 63

→ (x+ 13)

2 + (y − 23)

2 = 7

Mathematics 4 () Circles August 10, 2011 3 / 17

Review of Completing the Square

Completing the Square

Express the following expressions as (x− h)2 + (y − k)2 = r, where h, k,and r are constants.

1 x2 + y2 + 2x− 8y + 4 = 0

→ (x+ 1)2 + (y − 4)2 = 13

2 9x2 + 9y2 + 6x− 12y + 5 = 63

→ (x+ 13)

2 + (y − 23)

2 = 7

Mathematics 4 () Circles August 10, 2011 3 / 17

Review of Completing the Square

Completing the Square

Express the following expressions as (x− h)2 + (y − k)2 = r, where h, k,and r are constants.

1 x2 + y2 + 2x− 8y + 4 = 0

→ (x+ 1)2 + (y − 4)2 = 13

2 9x2 + 9y2 + 6x− 12y + 5 = 63

→ (x+ 13)

2 + (y − 23)

2 = 7

Mathematics 4 () Circles August 10, 2011 3 / 17

Circles

What is a circle?

Mathematics 4 () Circles August 10, 2011 4 / 17

Circles

Definition of Circles

A circle is a set of all points (locus) that are the same distance from agiven point.

Terminology

same distance → radius

given point → center

Mathematics 4 () Circles August 10, 2011 5 / 17

Circles

Definition of Circles

A circle is a set of all points (locus) that are the same distance from agiven point.

Terminology

same distance → radius

given point → center

Mathematics 4 () Circles August 10, 2011 5 / 17

Circles

Definition of Circles

A circle is a set of all points (locus) that are the same distance from agiven point.

Terminology

same distance → radius

given point → center

Mathematics 4 () Circles August 10, 2011 5 / 17

Circles

Definition of Circles

A circle is a set of all points (locus) that are the same distance from agiven point.

Terminology

same distance → radius

given point → center

Mathematics 4 () Circles August 10, 2011 5 / 17

The Standard Form of the Circle Equation

The Distance Formula

The distance between two points (x1, y1) and (x2, y2) in the Cartesianplane is given by:

d =√

(x2 − x1)2 + (y2 − y1)2

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The Standard Form of the Circle Equation

The Distance Formula

Use the distance formula to relate the radius with the center of the circle.

r =√(x− h)2 + (y − k)2 (1)

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The Standard Form of the Circle Equation

Standard Form/Center-Radius Form

Given a circle with center at (h, k) and having a radius r, the center radiusform of the circle equation is given by:

(x− h)2 + (y − k)2 = r2

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Graphing Examples

Graph x2 + y2 = 10. Label center, radius, and any intercepts.

x-intercepts →√10,−

√10

y-intercepts →√10,−

√10

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Graphing Examples

Graph x2 + y2 = 10. Label center, radius, and any intercepts.

x-intercepts →√10,−

√10

y-intercepts →√10,−

√10

Mathematics 4 () Circles August 10, 2011 9 / 17

Graphing Examples

Graph x2 + y2 = 10. Label center, radius, and any intercepts.

x-intercepts →√10,−

√10

y-intercepts →√10,−

√10

Mathematics 4 () Circles August 10, 2011 9 / 17

Graphing Examples

Graph (x+ 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.

x-intercepts → −3 +√5,−3−

√5

y-intercepts → 2

Mathematics 4 () Circles August 10, 2011 10 / 17

Graphing Examples

Graph (x+ 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.

x-intercepts → −3 +√5,−3−

√5

y-intercepts → 2

Mathematics 4 () Circles August 10, 2011 10 / 17

Graphing Examples

Graph (x+ 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.

x-intercepts → −3 +√5,−3−

√5

y-intercepts → 2

Mathematics 4 () Circles August 10, 2011 10 / 17

Problems on Circles

Example 1

What is the equation of a circle with radius 5, centered on the origin?Graph this circle.

x2 + y2 = 25

Mathematics 4 () Circles August 10, 2011 11 / 17

Problems on Circles

Example 1

What is the equation of a circle with radius 5, centered on the origin?Graph this circle.

x2 + y2 = 25

Mathematics 4 () Circles August 10, 2011 11 / 17

Problems on Circles

Example 1

What is the equation of a circle with radius 5, centered on the origin?Graph this circle.

x2 + y2 = 25

Mathematics 4 () Circles August 10, 2011 11 / 17

Problems on Circles

Example 1

What is the equation of a circle with radius 5, centered on the origin?Graph this circle.

x2 + y2 = 25

Mathematics 4 () Circles August 10, 2011 11 / 17

Problems on Circles

Example 2

Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts.

(x+ 3)2 + (y − 2)2 = 25

x-intercepts : −3±√21, y-intercepts: 6,−2

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Problems on Circles

Example 2

Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts.

(x+ 3)2 + (y − 2)2 = 25

x-intercepts : −3±√21, y-intercepts: 6,−2

Mathematics 4 () Circles August 10, 2011 12 / 17

Problems on Circles

Example 2

Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts.

(x+ 3)2 + (y − 2)2 = 25

x-intercepts : −3±√21, y-intercepts: 6,−2

Mathematics 4 () Circles August 10, 2011 12 / 17

Problems on Circles

Example 2

Move the circle in the previous problem 3 units to the left and 2 units up.What is its equation? Graph this circle and note the intercepts.

(x+ 3)2 + (y − 2)2 = 25

x-intercepts : −3±√21, y-intercepts: 6,−2

Mathematics 4 () Circles August 10, 2011 12 / 17

Problems on Circles

Example 3

Find an equation of a circle with a diameter whose endpoints are atP1(7,−3) and P2(1, 7).

center: Use the midpoint formula

→ (h, k) =

(x1 + x2

2,y1 + y2

2

)= (4, 2)

radius: Distance from one endpoint to the center→ r =

√(x1 − h)2 + (y1 − k)2 =

√34

(x− 4)2 + (y − 2)2 = 34

Mathematics 4 () Circles August 10, 2011 13 / 17

Problems on Circles

Example 3

Find an equation of a circle with a diameter whose endpoints are atP1(7,−3) and P2(1, 7).

center: Use the midpoint formula

→ (h, k) =

(x1 + x2

2,y1 + y2

2

)= (4, 2)

radius: Distance from one endpoint to the center→ r =

√(x1 − h)2 + (y1 − k)2 =

√34

(x− 4)2 + (y − 2)2 = 34

Mathematics 4 () Circles August 10, 2011 13 / 17

Problems on Circles

Example 3

Find an equation of a circle with a diameter whose endpoints are atP1(7,−3) and P2(1, 7).

center: Use the midpoint formula

→ (h, k) =

(x1 + x2

2,y1 + y2

2

)= (4, 2)

radius: Distance from one endpoint to the center→ r =

√(x1 − h)2 + (y1 − k)2 =

√34

(x− 4)2 + (y − 2)2 = 34

Mathematics 4 () Circles August 10, 2011 13 / 17

Problems on Circles

Example 3

Find an equation of a circle with a diameter whose endpoints are atP1(7,−3) and P2(1, 7).

center: Use the midpoint formula

→ (h, k) =

(x1 + x2

2,y1 + y2

2

)= (4, 2)

radius: Distance from one endpoint to the center→ r =

√(x1 − h)2 + (y1 − k)2 =

√34

(x− 4)2 + (y − 2)2 = 34

Mathematics 4 () Circles August 10, 2011 13 / 17

Problems on Circles

Example 3

Find an equation of a circle with a diameter whose endpoints are atP1(7,−3) and P2(1, 7).

center: Use the midpoint formula

→ (h, k) =

(x1 + x2

2,y1 + y2

2

)= (4, 2)

radius: Distance from one endpoint to the center→ r =

√(x1 − h)2 + (y1 − k)2 =

√34

(x− 4)2 + (y − 2)2 = 34

Mathematics 4 () Circles August 10, 2011 13 / 17

The General Form of the Circle Equation

Rewriting the answer to the previous problem:

(x− 4)2 + (y − 2)2 = 34

→ x2 + y2 − 8x− 4y − 14 = 0

This is called the General Form of the Circle Equation.

Mathematics 4 () Circles August 10, 2011 14 / 17

The General Form of the Circle Equation

Rewriting the answer to the previous problem:

(x− 4)2 + (y − 2)2 = 34 → x2 + y2 − 8x− 4y − 14 = 0

This is called the General Form of the Circle Equation.

Mathematics 4 () Circles August 10, 2011 14 / 17

The General Form of the Circle Equation

The General Form of the Circle Equation

Ax2 +Ay2 + Cx+Dy + E = 0

The x2 and y2 terms should have identical coefficients.

Mathematics 4 () Circles August 10, 2011 15 / 17

Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

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Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5

(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

Mathematics 4 () Circles August 10, 2011 16 / 17

Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

Mathematics 4 () Circles August 10, 2011 16 / 17

Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

Mathematics 4 () Circles August 10, 2011 16 / 17

Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

Mathematics 4 () Circles August 10, 2011 16 / 17

Problems on Circles

Example 4

Find the center and radius of the circle with the equationx2 + y2 + 4x− 6y + 5 = 0.

(x2 + 4x) + (y2 − 6y) = −5(x2 + 4x+4) + (y2 − 6y+9) = −5+4 + 9

(x+ 2)2 + (y − 3)2 = 8

C(−2, 3)r =√8 = 2

√2

Mathematics 4 () Circles August 10, 2011 16 / 17

Problems on Circles

More examples

1 Find the standard equation of a circle with center at (1, 5) and passesthrough (7, 2).

2 Find the area of the circle with equation x2 + y2 + 8x− 12y − 14.

3 Find the general equation of the circle tangent to both axes, whosecenter is in QII, and whose radius is 4.

Mathematics 4 () Circles August 10, 2011 17 / 17

Problems on Circles

More examples

1 Find the standard equation of a circle with center at (1, 5) and passesthrough (7, 2).

2 Find the area of the circle with equation x2 + y2 + 8x− 12y − 14.

3 Find the general equation of the circle tangent to both axes, whosecenter is in QII, and whose radius is 4.

Mathematics 4 () Circles August 10, 2011 17 / 17

Problems on Circles

More examples

1 Find the standard equation of a circle with center at (1, 5) and passesthrough (7, 2).

2 Find the area of the circle with equation x2 + y2 + 8x− 12y − 14.

3 Find the general equation of the circle tangent to both axes, whosecenter is in QII, and whose radius is 4.

Mathematics 4 () Circles August 10, 2011 17 / 17

Problems on Circles

More examples

1 Find the standard equation of a circle with center at (1, 5) and passesthrough (7, 2).

2 Find the area of the circle with equation x2 + y2 + 8x− 12y − 14.

3 Find the general equation of the circle tangent to both axes, whosecenter is in QII, and whose radius is 4.

Mathematics 4 () Circles August 10, 2011 17 / 17