Circle Shifts

Directions

1. Use the sliders to explore the parameters involved in determining the equation of a circle.[br][br]2. Answer the questions below.

Question 1

Describe the role of "r" in the equation of a circle.

Question 2

Describe the role of "h" in the equation of a circle.

Question 3

Describe the role of "k" in the equation of a circle.

Question 4

Set r = 5, C to (0,0) and A to (3,4). What do you notice about the right triangle associated with the radius and its vertical and horizontal components?[br][br]Once you have found this relationship, find at least 3 other "nice" points on this circle.

Question 5

Find another circle (change the radius and/or the center) and find a set of point that demonstrate the relationship you determined above.

Question 6

What is the equation of "Match My Function #1"?

[math]\frac{\left(x+2.5\right)^2}{3^2}+\frac{\left(y+4\right)^2}{3^2}=1[/math] [math]\frac{\left(x+2.5\right)^2}{3^2}+\frac{\left(y-4\right)^2}{3^2}=1[/math] [math]\frac{\left(x-2.5\right)^2}{3^2}+\frac{\left(y-4\right)^2}{3^2}=1[/math] [math]\frac{\left(x-2.5\right)^2}{3^2}+\frac{\left(y+4\right)^2}{3^2}=1[/math]

Question 7

What is the equation of "Match My Function #2"?

[math]\frac{\left(x+3\right)^2}{4}+\frac{\left(y-1.7\right)^2}{4}=1[/math] [math]\frac{\left(x+3\right)^2}{16}+\frac{\left(y-1.7\right)^2}{16}=1[/math] [math]\left(x+3\right)^2+\left(y-1.7\right)^2=4[/math] [math]\left(x+3\right)^2+\left(y-1.7\right)^2=1[/math]

Equation Formats

There are a number of ways to write the equation of a circle. For example:[br][math]\left(x-3\right)^2+\left(y+4\right)^2=25[/math] can also be written:[br][math]\frac{\left(x-3\right)^2}{6^2}+\frac{\left(y+4\right)^2}{6^2}=1[/math] to reflect that a circle is just a special ellipse. It can also be written:[br][math]x^2+y^2-6x+8y-11=0[/math] [br]Work with your group to make sure you understand how one might convert between these 3 forms of the equation before attempting the next set of questions.

Question 8

What is the radius of the circle described by the equation [math]\left(x-6\right)^2+\left(y+5\right)^2=49[/math]?

49 6 5 7

Question 9

What is the radius of the circle described by the equation [math]\frac{\left(x+5\right)^2}{9}+\frac{\left(y-4\right)^2}{9}=1[/math]?

4 5 3 9

Question 10

What is the center of the circle described by the equation [math]x^2+y^2-8x-14y+40=0[/math]?

(4 , -5) (4, -7) (4, 7) (-4 , 5)

Question 11

What is the radius of the circle described by the equation [math]x^2+y^2-8x-14y+40=0[/math]?

4 7 5 8 14 25