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Identifying a conic from its general form

Identify the graph of each of the following nondegenerate conic sections.

  1. 4 x 2 9 y 2 + 36 x + 36 y 125 = 0
  2. 9 y 2 + 16 x + 36 y 10 = 0
  3. 3 x 2 + 3 y 2 2 x 6 y 4 = 0
  4. 25 x 2 4 y 2 + 100 x + 16 y + 20 = 0
  1. Rewriting the general form, we have

    A = 4 and C = −9 , so we observe that A and C have opposite signs. The graph of this equation is a hyperbola.

  2. Rewriting the general form, we have

    A = 0 and C = 9. We can determine that the equation is a parabola, since A is zero.

  3. Rewriting the general form, we have

    A = 3 and C = 3. Because A = C , the graph of this equation is a circle.

  4. Rewriting the general form, we have

    A = −25 and C = −4. Because A C > 0 and A C , the graph of this equation is an ellipse.

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Identify the graph of each of the following nondegenerate conic sections.

  1. 16 y 2 x 2 + x 4 y 9 = 0
  2. 16 x 2 + 4 y 2 + 16 x + 49 y 81 = 0
  1. hyperbola
  2. ellipse
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Finding a new representation of the given equation after rotating through a given angle

Until now, we have looked at equations of conic sections without an x y term, which aligns the graphs with the x - and y -axes. When we add an x y term, we are rotating the conic about the origin. If the x - and y -axes are rotated through an angle, say θ , then every point on the plane may be thought of as having two representations: ( x , y ) on the Cartesian plane with the original x -axis and y -axis, and ( x , y ) on the new plane defined by the new, rotated axes, called the x' -axis and y' -axis. See [link] .

The graph of the rotated ellipse x 2 + y 2 x y 15 = 0

We will find the relationships between x and y on the Cartesian plane with x and y on the new rotated plane. See [link] .

The Cartesian plane with x - and y -axes and the resulting x ′− and y ′−axes formed by a rotation by an angle   θ .

The original coordinate x - and y -axes have unit vectors i and j . The rotated coordinate axes have unit vectors i and j . The angle θ is known as the angle of rotation    . See [link] . We may write the new unit vectors in terms of the original ones.

i = cos   θ i + sin   θ j j = sin   θ i + cos   θ j
Relationship between the old and new coordinate planes.

Consider a vector u in the new coordinate plane. It may be represented in terms of its coordinate axes.

u = x i + y j u = x ( i   cos   θ + j   sin   θ ) + y ( i   sin   θ + j   cos   θ ) Substitute . u = i x '   cos   θ + j x '   sin   θ i y '   sin   θ + j y '   cos   θ Distribute . u = i x '   cos   θ i y '   sin   θ + j x '   sin   θ + j y '   cos   θ Apply commutative property . u = ( x '   cos   θ y '   sin   θ ) i + ( x '   sin   θ + y '   cos   θ ) j Factor by grouping .

Because u = x i + y j , we have representations of x and y in terms of the new coordinate system.

x = x cos   θ y sin   θ and y = x sin   θ + y cos   θ

Equations of rotation

If a point ( x , y ) on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle θ from the positive x -axis, then the coordinates of the point with respect to the new axes are ( x , y ) . We can use the following equations of rotation to define the relationship between ( x , y ) and ( x , y ) :

x = x cos   θ y sin   θ

and

y = x sin   θ + y cos   θ

Given the equation of a conic, find a new representation after rotating through an angle.

  1. Find x and y where x = x cos   θ y sin   θ and y = x sin   θ + y cos   θ .
  2. Substitute the expression for x and y into in the given equation, then simplify.
  3. Write the equations with x and y in standard form.

Questions & Answers

given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
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Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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Abhi
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Abhi
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salma
Commplementary angles
Idrissa Reply
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Uday
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salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Practice Key Terms 3

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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