1. Circles! Circles! Circles!
9.3
Circles! Circles! Circles!

2. That’s One Way to Give Yourself a Headache

3. This is my friend the double-napped cone
This is my friend the double-napped cone. Say hello to the double-napped cone.

4. Let us slice my friend with a plane. What may result?

6. We’ll Study Those Four, but There Are Also 3 Degenerate Conics

7. Everybody Loves Circles
A circle is the set of points equidistant from a point C(h,k) called the center. The fixed distance r from the center to any point on the circle is called the radius.
The standard equation of a circle with center C(h,k) and radius r is as follows:
(x - h)2 + (y - k)2 = r2

8. Graph the Circle
(x-3)2 + y2 = 16 Center:_____(3,0) _______ Radius:______4______

9. Food for Thought x2+y2=0? (0,0) “point circle” x2+y2=-4?
Imaginary circles?? Not in Euclidean geometry.

10. Writing the Equation of a Circle
Center (1,-3) and Radius 6 __________________________
(x-1)2+(y+3)2=36

11. More Equations of Circles
Write in standard form and find the center and the radius: 1. Group x “stuff” and y “stuff. Move the constant to the other side. Get ready to complete the square: (x2-6x+__)+(y2-2y+__)=-4

12. Continued
2. Now complete the square. Don’t forget to add the stuff to the other side! (x2-6x+9)+(y2-2y+1)= Factor and simplify. (x-3)2+(y-1)2=6 This is a circle with center (3, 1) and radius

13. Practice Makes Perfect!
Write in standard form and graph.

14. Tangent Line to a Circle
Write the equation of the tangent line to the circle , x2+y2=13 at the point (2, 3).
We know the slope of the radius through the point (2,3) is 3/2 since the
circle is center at the origin.
Recall that a line tangent
to a circle is perpendicular
to the circle’s radius at the
point of tangency.

15. Continued
So just use the point (2,3) and the opposite reciprocal of the slope of the radius. So the tangent line will have slope of -2/3.
We get a tangent line of

16. You Try!
Find the equation of the circle with its center at (3, 5) tangent to the line x=-1.

17. Homework
Circles Worksheet

18. Homework Answers

19. More

20. More