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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.

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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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