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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

if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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