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What are conics?What are conics?

Conics, or conic sections, are the Conics, or conic sections, are the intersection of a plane with an infinite intersection of a plane with an infinite double cone. If that plane cuts both double cone. If that plane cuts both cones, it is a hyperbola. If it is parallel to cones, it is a hyperbola. If it is parallel to the edge of the cone, you get a parabola. the edge of the cone, you get a parabola. If neither is the case, it is an ellipse. The If neither is the case, it is an ellipse. The ellipse is also a circle if the plane is ellipse is also a circle if the plane is perpendicular to the altitude of the cone. perpendicular to the altitude of the cone.

CircleCircle

©National Science Foundation

CircleCircle

The Standard Form of a circle with a center at (0,0) and a The Standard Form of a circle with a center at (0,0) and a radius, r, is……..radius, r, is……..

222 ryx

center (0,0)radius = 2

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

CirclesCircles The Standard Form of a circle with a center at (h,k) and The Standard Form of a circle with a center at (h,k) and

a radius, r, is……..a radius, r, is……..

222 )()( rkyhx

center (3,3)radius = 2

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

Review: The geometric definition relies on a Review: The geometric definition relies on a cone and a plane intersecting itcone and a plane intersecting it

Algebraic definition: Algebraic definition: a set of points in the a set of points in the plane that are equidistant from a fixed point plane that are equidistant from a fixed point on the plane (the center).on the plane (the center).

Find the distance from the center of the Find the distance from the center of the circle (h,k) to any point on the circle circle (h,k) to any point on the circle (represented by (x,y)). This is the radius (represented by (x,y)). This is the radius of the circle.of the circle.

Review the distance formula:Review the distance formula:

Substitute in the values.Substitute in the values.

Square both sides to getSquare both sides to getthe general form of a the general form of a

circle in center-radius form.circle in center-radius form.

y

x

(h,k)

(x,y)

2 22 1 2 1( ) ( )d x x y y

2 2( ) ( )r x h y k r

2 2 2( ) ( )r x h y k

Radius (r)

Center (h,k)

Both variables are squared. Both variables are squared. Equation of a circle in center-radius form: Equation of a circle in center-radius form:

What makes the circle different from the What makes the circle different from the a line? a line?

What makes the circle different from the What makes the circle different from the parabola?parabola?

center is (3, 1) 4r

center is ( 5,2) 15r

center is (0,2) 3r

4. Write the equation of a circle centered 4. Write the equation of a circle centered at (2,-7) and having a radius of 5.at (2,-7) and having a radius of 5.

(x - 2)(x - 2)22 + (y + 7) + (y + 7)22 = 25 = 25

5. Describe (x - 2)5. Describe (x - 2)22 + (y + 1) + (y + 1)22 = 0 = 0

A point at (2,-1)A point at (2,-1)

6. Describe (x + 1)6. Describe (x + 1)22 + (y - 3) + (y - 3)22 = -1 = -1

No graphNo graph

7. Write the equation of a circle whose 7. Write the equation of a circle whose diameter is the line segment joining A(-3,-4) diameter is the line segment joining A(-3,-4) and B(4,3).and B(4,3).What must you find first?What must you find first?The center and the radius.The center and the radius.How can you find the center?How can you find the center?The center is the midpoint of the segment.The center is the midpoint of the segment.(½ , - ½ )(½ , - ½ )How can you find the radius?How can you find the radius?The radius is the distance from the center to a The radius is the distance from the center to a point on the circle. Use the distance formula.point on the circle. Use the distance formula.

The equation is: The equation is:

7 2r =

2

2 21 1 49

2 2 2x y

8. Write in center-radius form and 8. Write in center-radius form and sketch:sketch:

Hint: You must complete the Hint: You must complete the square.square.

2 26 ___ 4 ___ 12x x y y

2 2 6 4 12x y x y

2 26 9 4 4 12 9 4x x y y 2 2( 3) ( 2) 25x y

1)1) What’s the standard form of a line?What’s the standard form of a line?

2)2) What are the steps for graphing a What are the steps for graphing a circle?circle?

3)3) How can you tell if the graph of an How can you tell if the graph of an equation will be a line, a parabola, equation will be a line, a parabola, or a circle?or a circle?