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Identify the graph of each of the following nondegenerate conic sections.
$A=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=\mathrm{-9},$ so we observe that $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ have opposite signs. The graph of this equation is a hyperbola.
$A=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=9.\text{\hspace{0.17em}}$ We can determine that the equation is a parabola, since $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is zero.
$A=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=3.\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}A=C,$ the graph of this equation is a circle.
$A=\mathrm{-25}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=\mathrm{-4.}\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}AC>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}A\ne C,$ the graph of this equation is an ellipse.
Identify the graph of each of the following nondegenerate conic sections.
Until now, we have looked at equations of conic sections without an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term, which aligns the graphs with the x - and y -axes. When we add an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term, we are rotating the conic about the origin. If the x - and y -axes are rotated through an angle, say $\text{\hspace{0.17em}}\theta ,$ then every point on the plane may be thought of as having two representations: $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the Cartesian plane with the original x -axis and y -axis, and $\text{\hspace{0.17em}}\left({x}^{\prime},{y}^{\prime}\right)\text{\hspace{0.17em}}$ on the new plane defined by the new, rotated axes, called the x' -axis and y' -axis. See [link] .
We will find the relationships between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ on the Cartesian plane with $\text{\hspace{0.17em}}{x}^{\prime}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}^{\prime}\text{\hspace{0.17em}}$ on the new rotated plane. See [link] .
The original coordinate x - and y -axes have unit vectors $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}.$ The rotated coordinate axes have unit vectors $\text{\hspace{0.17em}}{i}^{\prime}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{j}^{\prime}.$ The angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is known as the angle of rotation . See [link] . We may write the new unit vectors in terms of the original ones.
Consider a vector $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because $\text{\hspace{0.17em}}u={x}^{\prime}{i}^{\prime}+{y}^{\prime}{j}^{\prime},$ we have representations of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in terms of the new coordinate system.
If a point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ from the positive x -axis, then the coordinates of the point with respect to the new axes are $\text{\hspace{0.17em}}\left({x}^{\prime},{y}^{\prime}\right).\text{\hspace{0.17em}}$ We can use the following equations of rotation to define the relationship between $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left({x}^{\prime},{y}^{\prime}\right):$
and
Given the equation of a conic, find a new representation after rotating through an angle.
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