circles the wheels on the bus go round and round (9.2)

Circles

The Wheels on the Bus Go Round and Round (9.2)

POD

Complete the square.

x2 + 6x – 7 = 0

POD

Complete the square.

x2 + 6x – 7 = 0 x2 + 6x = 7x2 + 6x + 9 = 7 + 9

(x + 3)(x + 3) = 16 (x + 3)2 = 42

What do you notice about this equation? What would its graph look like?

Today we start talking about conics

Conics are also referred to as “quadratic relations.”

The equations for these types of relations follow a certain kind of pattern.

Ax2 + Bxy + Cy2 +Dx + Ey + F = 0

Some of the coefficients could be zero. Then what happens to that term?

Today we start talking about conics

The first conic (or “conic section”) we’ll look at is circles.

What is the definition for a circle?

What are that point and that distance called?

Today we start talking about conics

The first conic (or “conic section”) we’ll look at is circles.

A circle:

The set of points on a plane equidistant from a given point.

What are that point and that distance called?

The center and the radius.

Circle equations—on the origin

If the center of the circle is on the origin, then the equation for the circle is given by

x2 + y2 = r2 where r is the radius.

Which coefficients equal 0 in the general equation?

Ax2 + Bxy + Cy2 +Dx + Ey + F = 0

Circle equations– on the origin

If the center of the circle is on the origin, then the equation for the circle is given by

x2 + y2 = r2 where r is the radius.

Sketch each of the following circles.1. x2 + y2 = 252. x2 + y2 = 493. x2 + y2 = 3

Circle equations– on the origin

Sketch each of the following circles.

1. x2 + y2 = 252. x2 + y2 = 493. x2 + y2 = 3

Circle equations– off the origin

If the center of the circle is off the origin, so that it’s at the point (h, k),

then the equation is given as

(x – h)2 + (y – k)2 = r2

Sketch each of the following circles.

1. (x – 2)2 + (y – 3)2 = 9

2. (x + 4)2 + (y – 1)2 = 93. x2 + (y+2)2 = 16

What do you need to know to graph them?

Circle equations– complete the square

When the square is obvious in the equation, it’s easy to find the center and radius. Sometimes we have to get the equation into that squared form.

That’s when we Complete the Square.

And sometimes we have to complete it for both the x and y variables.

Circle equations– complete the square

Complete the square and graph this circle.

x2 + y2 + 6x – 12 = 0

The Method:1. Move like terms in place and constants to

the right hand side.2. Complete the square(s).3. Factor completed square.

Circle equations– complete the square

1. Move like terms in place and constants to the right hand side.

x2 + 6x + y2 = 12

2. Complete the square(s).

x2 + 6x + 9 + y2 = 12 +9

3. Factor completed square.

(x + 3)2 + y2 = 21

Circle equations– complete the square

Now sketch it.

(x + 3)2 + y2 = 21

Circle equations– complete the square

Do the same with these equations.

x2 + y2 + 6x – 4y – 12 = 0

x2 + y2 – 10x +8y + 5 = 0

What terms from that general form for quadratic relations would be missing here?

Circle equations– complete the square

Do the same with these equations.

x2 + y2 + 6x – 4y – 12 = 0

(x + 3)2 + (y – 2)2 = 25

x2 + y2 – 10x +8y + 5 = 0

(x – 5)2 + (y + 4)2 = 36

Circle equations– complete the square

Sketch them.

(x + 3)2 + (y – 2)2 = 25

(x – 5)2 + (y + 4)2 = 36

Circle inequalities

What happens when the equal sign changes to an inequality?

Guess what these will look like.

x2 + y2 ≤ 25

x2 + y2 ≥ 25

Circle inequalities

What happens when the equal sign changes to an inequality?

Guess what these will look like.

x2 + y2 ≤ 25

x2 + y2 ≥ 25 One fills the inside and one

fills the outside.

Finally

Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2).

What information do you need?

How would you get it?

Finally

Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2).

What information do you need?The radius

How would you get it?The distance formula

Finally

Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2).

The distance formula:

65491674

)2(537

22

22

d

Finally

Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2).

The equation:

655)7( 22 yx

Finally

Find the equation for the circle with a center on (7, 5) and that contains the point (3, -2).

Another student approach:

65

4916

)7()4(

52)73(

2

222

222

r

r

r

r