Outline Distance and Midpoint Circles
College Algebra & Trigonometry I 2.8 - Distance and Midpoint Formulas; and Circles
Math 1100 North Carolina Central University Math & C.S. Department Hicham Qasmi -
[email protected]
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Distance and Midpoint Theorem
Math 1100
P2 (x2 , y2 ) •
d
• |x2 − x1 |
College Algebra & Trigonometry I
|y2 − y1 |
The distance d between two points (x1 , y1 ) and (x2 , y2 ) in the rectangular coordinate system is q • d = (x1 − x2 )2 + (y1 − y2 )2 P1 (x1 , y1 )
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Distance and Midpoint Theorem
Math 1100
P2 (x2 , y2 ) •
d
• |x2 − x1 |
College Algebra & Trigonometry I
|y2 − y1 |
The distance d between two points (x1 , y1 ) and (x2 , y2 ) in the rectangular coordinate system is q • d = (x1 − x2 )2 + (y1 − y2 )2 P1 (x1 , y1 )
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Example
Example Find the distance between (−1, 4) and (3, −2).
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Midpoint Formula Theorem
Consider the line segment from (x1 , y1 ) to (x2 , y2 ). The coordinates of the midpoint of the line segment are given by x + x y + y • 1 2 1 2 P1 (x1 , y1 ) , 2 2
Math 1100
P2 (x2 , y2 ) •
•
midpoint I
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Distance Example Midpoint Formula
Example
Example Find the midpoint of the line segment with endpoints (1, −6) and (−8, −4).
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Circles
Definition A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from the center to any point on the circle is called the radius.
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Standard Form of a Circle Equation Theorem The standard form of the equation of a circle with center (h, k) and radius r (with r > 0) is (x − h)2 + (y − k)2 = r 2 Proof. A point is on the circle if and only if its distance from the center (h, k) is equal to the radius r .In other words, one has p r = (x − h)2 + (y − k)2 r 2 = (x − h)2 + (y − k)2 after squaring both sides
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Standard Form of a Circle Equation Theorem The standard form of the equation of a circle with center (h, k) and radius r (with r > 0) is (x − h)2 + (y − k)2 = r 2 Proof. A point is on the circle if and only if its distance from the center (h, k) is equal to the radius r .In other words, one has p r = (x − h)2 + (y − k)2 r 2 = (x − h)2 + (y − k)2 after squaring both sides
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Standard Form of a Circle Equation Theorem The standard form of the equation of a circle with center (h, k) and radius r (with r > 0) is (x − h)2 + (y − k)2 = r 2 Proof. A point is on the circle if and only if its distance from the center (h, k) is equal to the radius r .In other words, one has p r = (x − h)2 + (y − k)2 r 2 = (x − h)2 + (y − k)2 after squaring both sides
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Standard Form of a Circle Equation Theorem The standard form of the equation of a circle with center (h, k) and radius r (with r > 0) is (x − h)2 + (y − k)2 = r 2 Proof. A point is on the circle if and only if its distance from the center (h, k) is equal to the radius r .In other words, one has p r = (x − h)2 + (y − k)2 r 2 = (x − h)2 + (y − k)2 after squaring both sides
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Standard Form of a Circle Equation Theorem The standard form of the equation of a circle with center (h, k) and radius r (with r > 0) is (x − h)2 + (y − k)2 = r 2 Proof. A point is on the circle if and only if its distance from the center (h, k) is equal to the radius r .In other words, one has p r = (x − h)2 + (y − k)2 r 2 = (x − h)2 + (y − k)2 after squaring both sides
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Example
Example 1
Write the standard form of the equation of the circle with center (0, 0) and radius 2.
2
Write the standard form of the equation of the circle with center (−2, 3) and radius 4.
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Recognize and Graph a Circle
Example 1
Find the center and radius of the circle whose equation is (x − 2)2 + (y + 4)2 = 9
2
Graph the equation.
3
Use the graph to identify the relations domain and range.
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Outline
1
Distance and Midpoint Distance Example Midpoint Formula
2
Circles Definition Equation of a Circle General Form of the Equation of a Circle
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
General Form of a Circle Equation
Theorem The general form of the equation of a circle is x 2 + y 2 + Dx + Ey + F = 0 where D, E and F are 3 real numbers
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation
To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so,
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation
To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation
To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side Example x 2 + y 2 + 4x − 6y − 23 = 0 + 4x ) + (y 2 − 6y ) = 23 (x 2 + 4x + 4) + (y 2 − 6y + 9) = 23 + 4 + 9 (x + 2)2 + (y − 3)2 = 36 (x 2
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side Example x 2 + y 2 + 4x − 6y − 23 = 0 + 4x ) + (y 2 − 6y ) = 23 (x 2 + 4x + 4) + (y 2 − 6y + 9) = 23 + 4 + 9 (x + 2)2 + (y − 3)2 = 36 (x 2
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side Example x 2 + y 2 + 4x − 6y − 23 = 0 + 4x ) + (y 2 − 6y ) = 23 (x 2 + 4x + 4) + (y 2 − 6y + 9) = 23 + 4 + 9 (x + 2)2 + (y − 3)2 = 36 (x 2
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side Example x 2 + y 2 + 4x − 6y − 23 = 0 + 4x ) + (y 2 − 6y ) = 23 (x 2 + 4x + 4) + (y 2 − 6y + 9) = 23 + 4 + 9 (x + 2)2 + (y − 3)2 = 36 (x 2
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Converting General Form to a Standard Form Equation To convert x 2 + y 2 + Dx + Ey + F = 0 into (x − h)2 + (y − k)2 = r 2 , one completes the square on x and y To do so, Arrange the x-terms in descending powers, as well as the y-terms Move the constant term to the right-hand-side Example x 2 + y 2 + 4x − 6y − 23 = 0 + 4x ) + (y 2 − 6y ) = 23 (x 2 + 4x + 4) + (y 2 − 6y + 9) = 23 + 4 + 9 (x + 2)2 + (y − 3)2 = 36 (x 2
Math 1100
College Algebra & Trigonometry I
Outline Distance and Midpoint Circles
Definition Equation of a Circle General Form of the Equation of a Circle
Now It’s Your Turn
Example Write in standard form and graph the circle x 2 + y 2 + 4x − 4y − 1 = 0
Math 1100
College Algebra & Trigonometry I
