10.3, Day 1
Circles! Circles! Circles!
Do Now: Take one of each of the 3
sheets
• Check your homework answers.
• 1) The diameter is about 11.93 cm.
• 2) The radius is about 58.56 m and the
velocity is about 20.04 m/s.
• Take a copy of the “Graphing Calculator Art
Project”…it will be due on 12/20…we will
discuss it more after you have a chance to
read it over.
That’s One Way to Give Yourself a
Headache
This is my friend the double-napped
cone. Say hello to the double-napped
cone.
Let us slice my friend with a plane.
What may result?
We’ll Study Those Four, but There Are
Also 3 Degenerate Conics
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
Graph the Circle
(x-3)2 + y2 = 16
Center:_____(3,0) _______
Radius:______4______
Food for Thought
• x2+y2=0?
(0,0) “point circle”
• x2+y2=-4?
• Imaginary circles?? Not in Euclidean geometry.
Writing the Equation of a Circle
• Center (1,-3) and Radius 6
__________________________
• (x-1)2+(y+3)2=36
More Equations of Circles
Write in standard form and find the center and
the radius: 2
2
x + y - 6x - 2y + 4 = 0
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
Continued
2. Now complete the square. Don’t forget to add
the stuff to the other side!
(x2-6x+9)+(y2-2y+1)=-4+9+1
3. Factor and simplify.
(x-3)2+(y-1)2=6
This is a circle with center (3, 1) and radius
6
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.
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
2
13
y=- x+
3
3
You Try!
• Find the equation of the circle with its center
at (3, 5) tangent to the line x=-1.
Practice Makes Perfect!
Write in standard form and graph.
3x +12x+3y -24y = -15
2
2