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P. H. Review 10.5-10.7 Conic Sections. C. E. We usually see conic equations written in General, or Implicit Form:. General Form of a Conic Equation. where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero. Please Note:.

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general form of a conic equation
We usually see conic equations written in General, or Implicit Form: General Form of a Conic Equation

where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.

please note
Please Note:

A conic equation written in General Form doesn’t have to have all SIX terms! Several of the coefficients A, B, C, D, E and F can equal zero, as long as A, B and C don’t ALL equal zero.

If A, B and C all equal zero, what kind of equation do you have?

...

T

H

I

N

K...

Linear!

so it s a conic equation if
So, it’s a conic equation if...
  • the highest degree (power) of x and/or y is 2 (at least ONE has to be squared)
  • the other terms are either linear, constant, or the product of x and y
  • there are no variable terms with rational exponents (i.e. no radical expressions) or terms with negative exponents (i.e. no rational expressions)
what values form an ellipse
The values of the coefficients in the conic equation determine the TYPE of conic.What values form an Ellipse?

What values form a Hyperbola?

What values form a Parabola?

ellipses
Ellipses...

where A & C have the SAME SIGN

NOTE: There is noBxy term, and D, E & Fmay equal zero!

For example:

ellipses1
This is an ellipse since x & y are both squared, and both quadratic terms have the same sign!

Center (-2, 0)

Hor. Axis = 2

Vert. Axis = √8

The General Form of the equations can be converted to Standard Form by completing the square and dividing so that the constant = 1.

Ellipses...
ellipses2
Vert. axis = 2/√3

center (-1, 1)

Hor. axis = 2

In this example, x2 and y2 are both negative (still the same sign), you can see in the final step that when we divide by negative 4 all of the terms are positive.

Ellipses...
ellipses a special case
Radius = √5

Center (-2, 0)

Ellipses…a special case!

When A & C are the samevalue as well as the same sign, the ellipse is the same length in all directions …

Circle!

it is a ...

hyperbola
Hyperbola...

where A & C have DIFFERENT signs.

NOTE: There is noBxy term, and D, E & Fmay equal zero!

For example:

hyperbola1
This is a hyperbola since x & y are both squared, and the quadratic terms have different signs!

y-axis=3

x-axis=2

Center (2,-1)

The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1.

Hyperbola...
hyperbola2
x-axis=2

Center (0,3)

y-axis=2

In this example, the signs change, but since the quadratic terms still have different signs, it is still a hyperbola!

Hyperbola...
parabola
A parabola is vertical if the equation has an x squared term AND a linear y term; it may or may not have a linear x term & constant:

A parabola is horizontal if the equation has a y squared term AND a linear x term; it may or may not have a linear y term & constant:

A Parabola can be oriented 2 different ways:

Parabola...
parabola vertical
Parabola …Vertical

The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term:

parabola vertical1
Vertex (2,3)Parabola …Vertical

To write the equations in Standard Form, complete the square for the x-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below:

parabola vertical2
Vertex (-1,4)Parabola …Vertical

In this example, the signs must be changed at the end so that the y-term is positive, notice that the negative coefficient of the x squared term makes the parabola open downward.

parabola horizontal
Parabola …Horizontal

The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term:

parabola horizontal1
Vertex (1,-4)Parabola …Horizontal

To write the equations in Standard Form, complete the square for the y-terms. There are 2 popular conventions for writing parabolas in Standard Form, both are given below:

parabola horizontal2
Vertex (3,0)Parabola …Horizontal

In this example, the signs must be changed at the end so that the x-term is positive; notice that the negative coefficient of the y squared term makes the parabola open to the left.

what about the term bxy
What About the term Bxy?

None of the conic equations we have looked at so far included the term Bxy. This term leads to a hyperbolic graph:

or, solved for y:

what about the term bxy1
If there is a Bxy term:What About the term Bxy?

You need to find the discriminant and use that to determine the conic section.

The graph is a circle (A = C) or an ellipse (A ≠ C)

The graph is a parabola

The graph is a hyperbola

summary
where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero.Summary ...

General Form of a Conic Equation:

Identifying a Conic Equation:

practice
Practice ...

Identify each of the following equations as a(n):

(a) ellipse (b) circle (c) hyperbola

(d) parabola (e) not a conic

See if you can rewrite each equation into its Graphing Form!

answers
Answers ...

(a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic

write the general from of the equation fo the translation of 6x 2 24x 4y 8 0 for t 1 2
Write the general from of the equation fo the translation of -6x2 + 24x + 4y – 8 = 0 for T(-1,-2)
solve this system of equations
Solve this system of equations:

Straight Line and a

circle

Not Factorable

NO SOLUTION!!!

Substitution:

Step 1

Solve for a variable

Step 2

Plug into other equation

solve this system of equations1
Solve this system of equations:

Hyperbola

And a straight line

Step 3

Plug into step 1 to find the other variable

Substitution:

Step 1

Solve for a variable

Step 2

Plug into other equation

Solution(s):

solve this system of equations2
Solve this system of equations:

CRICLE

And a ELLIPSE

Step 3

Plug into first equation to find the other variable

ELIMINATION:

Step 1

Make a new system

Step 2

Combine to eliminate

Solution(s):

solve this system of equations3
Solve this system of equations:

Hyperbola

And a Ellipse

Step 3

Plug into step 1 to find the other variable

Elimination:

Step 1

Re-write the system:

Step 2

Combine to eliminate

Solution(s):

solve this system of equations4
Solve this system of equations:

circle

And a straight line

Substitution:

Step 1

Solve for a variable

Step 3

Plug into step 1 to find the other variable

Step 2

Plug into other equation

Solution(s):

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