4.1 Circles The First Conic Section The basics, the definition, the formula And Tangent Lines

The Standard Form of a circle comes from the Distance Formula y1y1 x2x2 What is the length of the hypotenuse below? x1x1 y2y2

The Circle Definition; The set of all points in a plane a given distance _________________ from a given point ______________

The Circle The Formula ____________________________________________

Practice problems Find the center and radius of the following: Hint: Complete the Square

How to graph? This is probably the easiest part of it all. Essentially, plot the center then r And then label those 4 points. If r is irrational, then estimate its value and plot in the general correct area (and label with radical notation)

Notes 1._____________________________________ 2._____________________________________ ________________________________________ 1._____________________________________ ________________________________________

Graph

Now we will learn to find the equation of their tangent lines What do you know about a tangent line? _________________ So, if it perpendicular to the radius, what do you know about the slopes. ___________________________

What does a tangent line look like?

So, what are the steps? 1.________________________________ 2.________________________________ 3.________________________________ ________________________________ 4.________________________________ ________________________________ ________________________________ ________________________________ ________________________________

Find the equation of the line tangent to the circle at (2,2)

Group Problems: 2.Find the equation of the line tangent to the circle at (2,7). 3.Find the equation of the line tangent to the circle at the point in the 4 th quadrant where x = 4.

10-1 Ellipses Fumbles and Kickoffs

The Definition (Don’t write this down!!) The set of all points in a plane such that the sum of the distances from two fixed points, called foci (plural of focus) is a constant.

The Fumble

The Kickoff major axis b b a a (h,k)

So, how are we going to tell which is which? b a a b b b a a

That’s right!! a > b So, look for the larger number. If it is under x, _____________. If it is under the y, ______________________. When graphing, label center, major axis endpoints, minor axis endpoints and foci.

Foci, Where are they? They are on the major axis, “c” units from the center. How to find c?

Examples Graph the following completely. Remember: Standard Form = 1.

How to do this? Pull out any squared term coefficients before completing the square. Then divide so that the entire right side = 1.

10-1 Ellipses Graph information to Equation (going the other direction)

Lets just look at this standard form, the fumble, to see exactly what we need. You need the ______________________

Center You could be given ___________________________

a You could be given _____________________________

b You could be given ___________________________

Don’t forget! You also need the orientation!! I would TRULY suggest always doing a quick little sketch to see the axes orientation. A longer vertical axis is a A longer horizontal axis is a kickoff fumble

Examples 1.Center (1, 1); Focus (1, 3); Vertex (1, -9)

2. Foci (4,2) and (8, 2); MA endpoints (3, 2), (9, 2) Use MP formula to find Center: Use Fumble

10-2 Hyperbolas Day 1 Standard Equation and the Graph

The Definition The set of all points in a plane such that the difference of the distances from two points, called foci, is constant. Does that look familiar?

The Picture and Equation

The Other Orientation Happy/Sad

Day 1 is simply drawing the figure You need to draw the center, the tranverse axis endpoints (still a) and the asymptotes. We will use the “box method” (more later on that) to make sure that the shape is accurate. Do you remember what the relationship between a, b and c was in ellipses?

How to find foci This time, you take the sum of the denominators:

Lets go back to the definition: The set of all points in a plane such that the difference of the distances from two points, called foci, is constant. (h,k)

Remember: With ellipses, you move the number under each variable in that direction. It will be a very similar method with hyperbolas. Now is when we introduce the “Box Method”

The Box Method Move “a” units away from the center in both directions to form the transverse axis endpoints. Move “b” units away from the TA EP’s in both directions. _______________________________

10-2 Hyperbolas Graph information to Equation (going the other direction)

It’s the same process as before -How wonderful!! You need the _____________________________ Does this change with a happy/sad? So, how can we be given this information? (h,k)

Center You could be given ___________________________

a You could be given ____________________________

b You could be given _____________________________________ ________________________________________

Don’t forget! You still need the orientation!! I would again suggest doing a quick little sketch to see the orientation. A vertical traverse axis is a A horizontal traverse axis is a

Examples 1.Center (1, 3); TA ep (1, 7); Focus (1, -2)

2. TA eps (3, -3), (-5, -3); slope of asymptotes

10-3 Parabolas What? Again?

The definition The set of all points in a plane equidistant from a point (focus) and a line (directrix).

Parabola – “Up/Down”

“Left/Right”

So what do we do with this? 1._______________ and get it in conic form (but only if there is a linear term for x and/or y). 2.__________________________ 3.Put _________, __________, and _______ on the graph. Label these only. No tables or plotting 5 points this time.

Examples 1

Examples 1

6

4. Find the equation of the parabola with focus at (1, 5) and directrix at y =