Completing the Square Conic Section

Conic Sections

Circles

Ellipses

Parabolas

Hyperbolas

Poll

Do you remember how to find a radius and the center of a circle from center form?

• Yes

• No

• Kinda

Ellipses

A circle is a special kind of ellipse (Like a square is a special kind of rectangle)

An ellipse has a center, BUT it also has 2 FOCI - the are kind of like centers, but pulled out along one direction, stretching the ellipse so that it is longer on one side.

Circle to Ellipse

• Top - Circle - the Foci and the Center are all the same

• Middle - Horizontal Ellipse - the foci have been moved out and are no longer the same as the center

• Bottom - Horizontal Ellipse - foci are even further from the center

Open Ended

If I gave you the circle equation

(x - 3)² + (y + 5)² = 36,

how would you set it to = 1?

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This says that the center of the circle is (3, -5) with a radius of 6 because all I did to the original equation is divide everything by 36.  This is a CIRCLE with a radius of 6.

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This is no longer a circle, it is an ELLIPSE.  This does not have a single radius of 6 anymore.  Now it has many different radii, including a MAJOR radius and a MINOR radius.

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Poll

Which way do you think the major radius of

$$\frac{\left(x-3\right)^2}{9}\ +\ \frac{\left(y+5\right)^2}{36}\ =\ 1$$  will go when you graph it?

• Horizontally

• Vertically

Poll

Which way do you think the minor radius of

$$\frac{\left(x-3\right)^2}{9}\ +\ \frac{\left(y+5\right)^2}{36}\ =\ 1$$  will go when you graph it?

• Horizontally

• Vertically

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This is a VERTICAL ELLIPSE (It is taller than it is wide) A HORIZONTAL ELLIPSE is wider than it is tall 

Multiple Choice

How will you know if an ellipse is vertical?

• The denominators will all be the same

• The bigger denominator will be under the x

• The bigger denominator will be under the y

Graph

Multiple Choice

How is the graph of

$$\frac{\left(x-3\right)^2}{36}\ +\ \frac{\left(y+5\right)^2}{9}\ =\ 1$$  different?

• It's not

• It's vertical

• It's a circle

• It's horizontal

Graph

Multiple Select

Check all the boxes that are true about the following ellipse

$$\frac{x^2}{25}+\frac{\left(y-2\right)^2}{16}=1$$  

• Center (1, 2)

• Center (0, 2)

• Horizontal Ellipse

• Vertical Ellipse

• Major radius of 5

Multiple Choice

Which is the graph of

$$\frac{x^2}{25}+\frac{\left(y-2\right)^2}{16}=1$$  

Multiple Select

What is true about

$$\frac{\left(x+4\right)^2}{9}\ +\ \frac{\left(y-1\right)^2}{25}\ =\ 1$$  

• Center (-4,1)

• Center (4,-1)

• Center (3,5)

• Center (9,25)

Multiple Select

What is true about

$$\frac{\left(x+4\right)^2}{9}\ +\ \frac{\left(y-1\right)^2}{25}\ =\ 1$$  

• Major radius 3 minor 5

• Major radius 5 minor 3

• Horizontal Ellipse

• Vertical Ellipse

Poll

How are you feeling about this so far? 1 - No Clue to 5 - I Got it

• 1

• 2

• 3

• 4

• 5