1. DEFINITION OF A CIRCLE and example
Standard 4, 9, 17
CIRCLES
DEFINITION OF A CIRCLE and example
PROBLEM 1a
PROBLEM 1b
PROBLEM 2a
PROBLEM 2b
PROBLEM 3
END SHOW
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2. ALGEBRA II STANDARDS THIS LESSON AIMS:
Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes
ESTÁNDAR 4:
Los estudiantes factorizan polinomios representando diferencia de cuadrados, trinomios cuadrados perfectos, y la suma de diferencia de cubos.
STANDARD 9:
Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) + c.
ESTÁNDAR 9:
Los estudiantes demuestran y explican los efectos que tiene el cambiar coeficientes en la gráfica de funciones cuadráticas; esto es, los estudiantes determinan como la gráfica de una parabola cambia con a, b, y c variando en la ecuación y=a(x-b) + c
STANDARD 17:
Given a quadratic equation of the form ax + by + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
Estándar 17:
Dada una equación cuadrática de la forma ax +by + cx + dy + e=0, los estudiantes pueden usar el método de completar al cuadrado para poner la ecuación en forma estándar y pueden reconocer si la gráfica es un círculo, elipse, parábola o hiperbola. Los estudiantes pueden graficar la ecuación 2
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3. Definition of a Circle:
Standard 4, 9, 17
CIRCLES
Definition of a Circle:
A circle is the set of all points in a plane that are equidistant from a given point in the plane, called the center. Any segment whose endpoints are the center and a point on the circle is a radius of the circle.
Equation of a Circle:
The equation of a circle with center at (h,k) and radius r units is
(x – h) + (y – k) = r 2 4 2 6 -2 -4 -6 8 10 -8
-10 x y
What would be the equation for this circle?
h = 3
k = 2
(3,2) 6
r = 6
(x – ) + (y – ) = ( ) 2 3 2 6
(x – ) + (y – ) = 36 2 3
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4. Standard 4, 9, 17
Write the equation of a circle whose center is at (5, -2) and the circle passes through the point (2, 3). 4 2 6 -2 -4 -6 8 10 -8
-10 x y y 1 x
=(2,3)
(2,3) y 2 x
=(5,-2) 34
r = (x –x ) + (y –y ) 2 1
(5,-2)
r = ( ) + ( ) 2 5 2 -2 3
= ( 3 ) + ( -5 ) 2
(x – h) + (y – k) = r 2 =
(x – ) + (y – ) = ( ) 2 5 -2 34
r= 34
(x – ) + (y + ) = 34 2 5
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5. Standard 4, 9, 17
Write the equation of a circle whose center is at (6, -4) and the circle passes through the point (3, 4). 4 2 6 -2 -4 -6 8 10 -8
-10 x y y 1 x
=(3,4)
(3,4) y 2 x
=(6,-4) 73
r = (x –x ) + (y –y ) 2 1
(6,-4)
r = ( ) + ( ) 2 6 3 -4 4
= ( 3 ) + ( -8 ) 2
(x – h) + (y – k) = r 2 =
(x – ) + (y – ) = ( ) 2 6 -4 73
r= 73
(x – ) + (y + ) = 73 2 6 4
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6. and then graph the corresponding circle. (x – h) + (y – k) = r
Standard 4, 9, 17
Given that x + y + 8x +2y -32=0 is the equation of a circle. Put it in the form
and then graph the corresponding circle.
(x – h) + (y – k) = r 2
Changing the form:
Rewriting the equation to graph it:
(x –( )) + (y –( )) = ( ) 2 -4 -1 7
x + y + 8x +2y -32=0 2
(x – h) + (y – k) = r 2 8 2 2
x + 8x y + 2y = 2
h= -4
k= -1
r= 7
(4) 2
(1) 2 4 2 6 -2 -4 -6 8 10 -8
-10 x y
x + 8x y + 2y = 2
x + 8x y + 2y = 2 16 1 7
(x + 4) 2
(y + 1) 2 +
- 32 = 17
(-4,-1)
(x + ) + (y + ) = 49 2 4 1
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7. and then graph the corresponding circle. (x – h) + (y – k) = r
Standard 4, 9, 17
Given that x + y + 6x +4y -23=0 is the equation of a circle. Put it in the form
and then graph the corresponding circle.
(x – h) + (y – k) = r 2
Changing the form:
Rewriting the equation to graph it:
(x –( )) + (y –( )) = ( ) 2 -3 -2 6
x + y + 6x +4y -23=0 2
(x – h) + (y – k) = r 2 6 2 4 2
x + 6x y + 4y = 2
h= -3
k= -2
r= 6
(3) 2
(2) 2 4 2 6 -2 -4 -6 8 10 -8
-10 x y
x + 6x y + 4y = 2
x + 6x y + 4y = 2 9 4
(x + 3) 2
(y + 2) 2 6 +
- 23 = 13
(-3,-2)
(x + ) + (y + ) = 36 2 3
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8. First we find the midpoint or center of the circle:
Standard 4, 9, 17
Write an equation of a circle if the endpoints of a diameter are at (-1,-3), and (7,5). x 2 -1 y 2 -3 4 2 6 -2 -4 -6 8 10 -8
-10 x y x 1 7 y 1 5
First we find the midpoint or center of the circle: x 1 2 , + y 1 2 + =
Using: x, y
(3,1)
4 2 = , 2 + x, y = y x, 2 6 , = y x, 1 3,
Center of the circle
r = (x –x ) + (y –y ) 2 1
Now we calculate the radius’ length using: 32 2 2
r = ( ) + ( ) 2 3 7 1 5 y 1 x
=(7,5) 16 2 8 2 2 4 2
= ( -4) + (-4) 2 y 2 x
=(3,1) 2 2 1 = = 2 2 2 2 32 = =
=4 2
r= 32
=4 2
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9. Standard 4, 9, 17
Now using the coordinates found for the center and the radius to find the equation for the circle: 4 2 6 -2 -4 -6 8 10 -8
-10 x y
4 2
(3,1)
h = 3
k = 1
r =
4 2
r= 32
r = 32 2
(x – h) + (y – k) = r 2
(x – ) + (y – ) = ( ) 2 3 1
4 2
(x – ) + (y – ) = 32 2 3 1
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