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10.4 Rotation of axes

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In this section, you will:
  • Identify nondegenerate conic sections given their general form equations.
  • Use rotation of axes formulas.
  • Write equations of rotated conics in standard form.
  • Identify conics without rotating axes.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone . The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See [link] .

The nondegenerate conic sections

Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections , in contrast to the degenerate conic sections    , which are shown in [link] . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.

Degenerate conic sections

Identifying nondegenerate conics in general form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.

A x 2 + B x y + C y 2 + D x + E y + F = 0

where   A , B , and   C   are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.

You may notice that the general form equation has an   x y   term that we have not seen in any of the standard form equations. As we will discuss later, the   x y   term rotates the conic whenever   B   is not equal to zero.

Conic Sections Example
ellipse 4 x 2 + 9 y 2 = 1
circle 4 x 2 + 4 y 2 = 1
hyperbola 4 x 2 − 9 y 2 = 1
parabola 4 x 2 = 9 y  or  4 y 2 = 9 x
one line 4 x + 9 y = 1
intersecting lines ( x − 4 ) ( y + 4 ) = 0
parallel lines ( x − 4 ) ( x − 9 ) = 0
a point 4 x 2 + 4 y 2 = 0
no graph 4 x 2 + 4 y 2 =   −   1

General form of conic sections

A conic section    has the general form

A x 2 + B x y + C y 2 + D x + E y + F = 0

where   A , B , and   C   are not all zero.

[link] summarizes the different conic sections where   B = 0 , and   A   and   C   are nonzero real numbers. This indicates that the conic has not been rotated.

ellipse A x 2 + C y 2 + D x + E y + F = 0 ,   A ≠ C  and  A C > 0
circle A x 2 + C y 2 + D x + E y + F = 0 ,   A = C
hyperbola A x 2 − C y 2 + D x + E y + F = 0  or  − A x 2 + C y 2 + D x + E y + F = 0 , where   A   and   C   are positive
parabola A x 2 + D x + E y + F = 0  or  C y 2 + D x + E y + F = 0

Given the equation of a conic, identify the type of conic.

  1. Rewrite the equation in the general form, A x 2 + B x y + C y 2 + D x + E y + F = 0.
  2. Identify the values of   A   and   C   from the general form.
    1. If   A   and   C   are nonzero, have the same sign, and are not equal to each other, then the graph may be an ellipse.
    2. If   A   and   C   are equal and nonzero and have the same sign, then the graph may be a circle.
    3. If   A   and   C   are nonzero and have opposite signs, then the graph may be a hyperbola.
    4. If either   A   or   C   is zero, then the graph may be a parabola.

    If B = 0, the conic section will have a vertical and/or horizontal axes. If B does not equal 0, as shown below, the conic section is rotated. Notice the phrase “may be” in the definitions. That is because the equation may not represent a conic section at all, depending on the values of A , B , C , D , E , and F . For example, the degenerate case of a circle or an ellipse is a point:
    A x 2 + B y 2 = 0 , when A and B have the same sign.
    The degenerate case of a hyperbola is two intersecting straight lines: A x 2 + B y 2 = 0 , when A and B have opposite signs.
    On the other hand, the equation, A x 2 + B y 2 + 1 = 0 , when A and B are positive does not represent a graph at all, since there are no real ordered pairs which satisfy it.

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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