Section 1.4

Circles

Math 1051 - Precalculus I

Circles Section 1.4

Section 1.4 Circles

Find the equation of the line that passes through (2,−3) and(1,−2)

Circles Section 1.4

From last time...

What can you say about 2 lines that have the same x-interceptand the same y -intercept?

Circles Section 1.4

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Circles Section 1.4

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Circles Section 1.4

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Circles Section 1.4

Circle

How to describe a circle:

Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circle

How to describe a circle:Start with a fixed point (h, k)

Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circle

How to describe a circle:Start with a fixed point (h, k)Give a distance r

A circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circle

How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circle

How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circle

How to describe a circle:Start with a fixed point (h, k)Give a distance rA circle is all the points in the xy -plane that are a distancer from the point (h, k)

Definitions:r is the radius(h, k) is the center

Circles Section 1.4

Circles Section 1.4

Facts about circles

A = πr2

c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Facts about circles

A = πr2

c = 2πr

π ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Facts about circles

A = πr2

c = 2πrπ ≈ 3.1415926535897932384626433832795

π is an irrational numberπ is a transcendental numberBut

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Facts about circles

A = πr2

c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational number

π is a transcendental numberBut

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Facts about circles

A = πr2

c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental number

But

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Facts about circles

A = πr2

c = 2πrπ ≈ 3.1415926535897932384626433832795π is an irrational numberπ is a transcendental numberBut

π = 4∞∑

k=0

(−1)k

2k + 1=

41− 4

3+

45− 4

7+ · · ·

Circles Section 1.4

Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits

1760280

≈ 2π

Circles Section 1.4

Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits

1760280

≈ 2π

Circles Section 1.4

Great Pyramid of GizaHas a perimeter of 1760 cubits and a height of 280 cubits

1760280

≈ 2π

Circles Section 1.4

Formula for a circle

Standard form

(x − h)2 + (y − k)2 = r2

General form

x2 + y2 + ax + by + c = 0

Circles Section 1.4

Formula for a circle

Standard form

(x − h)2 + (y − k)2 = r2

General form

x2 + y2 + ax + by + c = 0

Circles Section 1.4

Formula for a circle

Standard form

(x − h)2 + (y − k)2 = r2

General form

x2 + y2 + ax + by + c = 0

Circles Section 1.4

Examples

Find the center, radius, and intercepts of

3(x + 1)2 + 3(y − 1)2 = 6

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Circles Section 1.4

Examples

Find the center, radius, and intercepts of

3(x + 1)2 + 3(y − 1)2 = 6

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Circles Section 1.4

Examples

Find the center, radius, and intercepts of

x2 + y2 + 4x + 2y − 20 = 0

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Circles Section 1.4

Examples

Find the center, radius, and intercepts of

x2 + y2 + 4x + 2y − 20 = 0

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Circles Section 1.4

Examples

Find the center, radius and intercepts of

2x2 + 2y2 + 8x + 7 = 0

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Circles Section 1.4

Examples

Find the center, radius and intercepts of

2x2 + 2y2 + 8x + 7 = 0

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4

Circles Section 1.4

Examples

Find the equation of a circle with endpoints of a diameter (4,3)and (0,1)

Important: The information we need for a circle is the centerand radius

Circles Section 1.4

Examples

Find the equation of a circle with endpoints of a diameter (4,3)and (0,1)

Important: The information we need for a circle is the centerand radius

Circles Section 1.4

Examples

Find the equation of a circle with center (4,−2) and tangent tothe line x = 1.

Circles Section 1.4

Examples

Find the area of the region between the circle x2 + y2 = 36 andthe inscribed square.

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Circles Section 1.4

Examples

Find the area of the region between the circle x2 + y2 = 36 andthe inscribed square.

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Circles Section 1.4

Examples

Given the circle x2 + y2 = r2 and a line y = mx + b tangent tothe circle, find b as a function of r and m.

Circles Section 1.4

Read sections 2.1 and 2.2 for Monday

Circles Section 1.4