Until now, we have looked at equations of conic sections without an \(xy\) term, which aligns the graphs with the x- and y-axes. When we add an \(xy\) term, we are rotating the conic about the origin. If the x- and y-axes are rotated through an angle, say \(\theta\),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 \((x^\prime ,y^\prime )\) on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis (Figure \(\PageIndex{3}\)). Show  Figure \(\PageIndex{3}\): The graph of the rotated ellipse \(x^2+y^2–xy–15=0\) We will find the relationships between \(x\) and \(y\) on the Cartesian plane with \(x^\prime \) and \(y^\prime \) on the new rotated plane (Figure \(\PageIndex{4}\)). Figure \(\PageIndex{4}\): The Cartesian plane with \(x\)- and \(y\)-axes and the resulting \(x^\prime\)− and \(y^\prime\)−axes formed by a rotation by an angle \(\theta\). The original coordinate x- and y-axes have unit vectors \(\hat{i}\) and \(\hat{j}\). The rotated coordinate axes have unit vectors \(\hat{i}^\prime\) and \(\hat{j}^\prime\).The angle \(\theta\) is known as the angle of rotation (Figure \(\PageIndex{5}\)). We may write the new unit vectors in terms of the original ones. \[\hat{i}′=\cos \theta \hat{i}+\sin \theta \hat{j}\] \[\hat{j}′=−\sin \theta \hat{i}+\cos \theta \hat{j}\] Figure \(\PageIndex{5}\): Relationship between the old and new coordinate planes. Consider a vector \(\vec{u}\) in the new coordinate plane. It may be represented in terms of its coordinate axes. \[\begin{align*} \vec{u}&=x^\prime i′+y^\prime j′ \\[4pt] &=x^\prime (i \cos \theta+j \sin \theta)+y^\prime (−i \sin \theta+j \cos \theta) & \text{Substitute.} \\[4pt] &=ix' \cos \theta+jx' \sin \theta−iy' \sin \theta+jy' \cos \theta & \text{Distribute.} \\[4pt] &=ix' \cos \theta−iy' \sin \theta+jx' \sin \theta+jy' \cos \theta & \text{Apply commutative property.} \\[4pt] &=(x' \cos \theta−y' \sin \theta)i+(x' \sin \theta+y' \cos \theta)j & \text{Factor by grouping.} \end{align*}\] Because \(\vec{u}=x^\prime i′+y^\prime j′\), we have representations of \(x\) and \(y\) in terms of the new coordinate system. \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\)
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 \(\theta\) from the positive x-axis, then the coordinates of the point with respect to the new axes are \((x^\prime ,y^\prime )\). We can use the following equations of rotation to define the relationship between \((x,y)\) and \((x^\prime , y^\prime )\): \[x=x^\prime \cos \theta−y^\prime \sin \theta\] and \[y=x^\prime \sin \theta+y^\prime \cos \theta\]
Find a new representation of the equation \(2x^2−xy+2y^2−30=0\) after rotating through an angle of \(\theta=45°\). Solution Find \(x\) and \(y\), where \(x=x^\prime \cos \theta−y^\prime \sin \theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\). Because \(\theta=45°\), \[\begin{align*} x &= x^\prime \cos(45°)−y^\prime \sin(45°) \\[4pt] x &= x^\prime \left(\dfrac{1}{\sqrt{2}}\right)−y^\prime \left(\dfrac{1}{\sqrt{2}}\right) \\[4pt] x &=\dfrac{x^\prime −y^\prime }{\sqrt{2}} \end{align*}\] and \[\begin{align*} y &= x^\prime \sin(45°)+y^\prime \cos(45°) \\[4pt] y &= x^\prime \left(\dfrac{1}{\sqrt{2}}\right) + y^\prime \left(\dfrac{1}{\sqrt{2}}\right) \\[4pt] y &= \dfrac{x^\prime +y^\prime }{\sqrt{2}} \end{align*}\] Substitute \(x=x^\prime \cos\theta−y^\prime \sin\theta\) and \(y=x^\prime \sin \theta+y^\prime \cos \theta\) into \(2x^2−xy+2y^2−30=0\). \(2{\left(\dfrac{x^\prime −y^\prime }{\sqrt{2}}\right)}^2−\left(\dfrac{x^\prime −y^\prime }{\sqrt{2}}\right)\left(\dfrac{x^\prime +y^\prime }{\sqrt{2}}\right)+2{\left(\dfrac{x^\prime +y^\prime }{\sqrt{2}}\right)}^2−30=0\) Simplify. \(\begin{array}{rl} 2\dfrac{(x^\prime−y^\prime )(x^\prime −y^\prime )}{2}−\dfrac{(x^\prime −y^\prime )(x^\prime +y^\prime )}{2}+2\dfrac{(x^\prime +y^\prime )(x^\prime +y^\prime )}{2}−30=0 & \text{FOIL method} \\[4pt] {x^\prime }^2−2x^\prime y^\prime +{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2}+{x^\prime }^2+2x^\prime y^\prime +{y^\prime }^2−30=0 & \text{Combine like terms.} \\[4pt] 2{x^\prime }^2+2{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2}=30 & \text{Combine like terms.} \\[4pt] 2(2{x^\prime }^2+2{y^\prime }^2−\dfrac{({x^\prime }^2−{y^\prime }^2)}{2})=2(30) & \text{Multiply both sides by 2.} \\[4pt] 4{x^\prime }^2+4{y^\prime }^2−({x^\prime }^2−{y^\prime }^2)=60 & \text{Simplify. } \\[4pt] 4{x^\prime }^2+4{y^\prime }^2−{x^\prime }^2+{y^\prime }2=60 & \text{Distribute.} \\[4pt] \dfrac{3{x^\prime }^2}{60}+\dfrac{5{y^\prime }^2}{60}=\dfrac{60}{60} & \text{Set equal to 1.} \end{array} \) Write the equations with \(x^\prime \) and \(y^\prime \) in the standard form. \[\dfrac{{x^\prime }^2}{20}+\dfrac{{y^\prime}^2}{12}=1 \nonumber\] This equation is an ellipse. Figure \(\PageIndex{6}\) shows the graph. Figure \(\PageIndex{6}\) A conic section is any of the geometric figures that can arise when a plane intersects a cone. (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves, and every conic section can be described in Cartesian coordinates by a polynomial equation of degree two or less. Classification of conic sectionsAll conic sections fall into the following categories: Nondegenerate conic sections
Degenerate conic sectionsIf the cutting plane passes through the vertex of the cone, the result is a degenerate conic section. Degenerate conics fall into three categories:
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Definitions of conic sections in terms of foci and directrices
Definitions of conic sections in terms of Cartesian coordinates
See Also
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What happens to the parabola when it degenerates
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