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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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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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