Download - Circles Lecture - Part 1
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
Mathematics 4 () Circles August 10, 2011 6 / 17
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)
Mathematics 4 () Circles August 10, 2011 7 / 17
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
Mathematics 4 () Circles August 10, 2011 8 / 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 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
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 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
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
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