Graph the following equations.
\(x^2+2xy+y^2 -x\sqrt {2}+y\sqrt {2} -6= 0\)

\(x^2+2xy+y^2 -x\sqrt {2}+y\sqrt {2} -6= 0\) becomes \((x')^2 = -(y'-3)\) after rotating counter-clockwise through \(\theta = \frac {\pi }{4}\).

[Picture]

\(7x^2-4xy\sqrt {3}+3y^2-2x-2y\sqrt {3}-5= 0\)

\(7x^2-4xy\sqrt {3}+3y^2-2x-2y\sqrt {3}-5= 0\) becomes \(\frac {(x'-2)^2}{9}+(y')^2 = 1\) after rotating counter-clockwise through \(\theta = \frac {\pi }{3}\)

[Picture]

\(5x^2+6xy+5y^2 - 4\sqrt {2}x+4\sqrt {2}y = 0\)

\(5x^2+6xy+5y^2 - 4\sqrt {2}x+4\sqrt {2}y = 0\) becomes \((x')^2+\frac {(y'+2)^2}{4} = 1\) after rotating counter-clockwise through \(\theta = \frac {\pi }{4}\).

[Picture]

\(x^2+ 2\sqrt {3}xy+3y^2+ 2\sqrt {3}x-2y-16 = 0\)

\(x^2+ 2\sqrt {3}xy+3y^2+ 2\sqrt {3}x-2y-16 = 0\) becomes\((x')^2 = y'+4\) after rotating counter-clockwise through \(\theta = \frac {\pi }{3}\)

[Picture]

\(13x^2-34xy\sqrt {3}+47y^2 - 64=0\)

\(13x^2-34xy\sqrt {3}+47y^2 - 64=0\) becomes \((y')^2 - \frac {(x')^2}{16} =1 \) after rotating counter-clockwise through \(\theta = \frac {\pi }{6}\).

[Picture]

\(x^2-2\sqrt {3} xy-y^2+8=0\)

\(x^2-2\sqrt {3} xy-y^2+8=0\) becomes \(\frac {(x')^2}{4} - \frac {(y')^2}{4} = 1\) after rotating counter-clockwise through \(\theta = \frac {\pi }{3}\)

[Picture]

\(x^2-4xy+4y^2-2x\sqrt {5}-y\sqrt {5}=0\)

\(x^2-4xy+4y^2-2x\sqrt {5}-y\sqrt {5}=0\) becomes \((y')^2=x\) after rotating counter-clockwise through \(\theta = \arctan \left (\frac {1}{2}\right )\). [Picture]

\(8x^2+12xy+17y^2 - 20 = 0\)

\(8x^2+12xy+17y^2 - 20 = 0\) becomes \((x')^2 + \frac {(y')^2}{4} = 1\) after rotating counter-clockwise through \(\theta = \arctan (2)\).

[Picture]

Graph the following equations.
\(r = \frac {2}{1-\cos (\theta )}\)

\(r = \frac {2}{1-\cos (\theta )}\) is a parabola

directrix: \(x = -2\)

vertex: \((-1,0)\)

focus: \((0,0)\)

focal diameter: \(4\)

[Picture]

\(r = \frac {3}{2 + \sin (\theta )}\)

\(r = \frac {3}{2 + \sin (\theta )} = \frac {\frac {3}{2}}{1 + \frac {1}{2} \sin (\theta )}\) is an ellipse

directrix: \(y = 3\)

vertices: \((0,1)\), \((0,-3)\)

center: \((0,-2)\)

foci: \((0,0)\), \((0,-2)\)

minor axis length: \(2\sqrt {3}\)

[Picture]

\(r = \frac {3}{2-\cos (\theta )}\)

\(r = \frac {3}{2 - \cos (\theta )} = \frac {\frac {3}{2}}{1 - \frac {1}{2} \cos (\theta )}\) is an ellipse

directrix: \(x = -3\)

vertices: \((-1,0)\), \((3,0)\)

center: \((1,0)\)

foci: \((0,0)\), \((2,0)\)

minor axis length: \(2\sqrt {3}\)

[Picture]

\(r = \frac {2}{1 + \sin (\theta )}\)

\(r = \frac {2}{1+\sin (\theta )}\) is a parabola

directrix: \(y=2\)

vertex: \((0,1)\)

focus: \((0,0)\)

focal diameter: \(4\)

[Picture]

\(r = \frac {4}{1+3\cos (\theta )}\)

\(r = \frac {4}{1+3\cos (\theta )}\) is a hyperbola

directrix: \(x = \frac {4}{3}\)

vertices: \((1,0)\), \((2,0)\)

center: \(\left (\frac {3}{2}, 0\right )\)

foci: \((0,0)\), \((3,0)\)

conjugate axis length: \(2\sqrt {2}\)

[Picture]

\(r = \frac {2}{1-2\sin (\theta )}\)

\(r = \frac {2}{1-2\sin (\theta )}\) is a hyperbola

directrix: \(y = -1\)

vertices: \(\left (0,-\frac {2}{3}\right )\), \((0,-2)\)

center:\(\left (0, -\frac {4}{3} \right )\)

foci: \((0,0)\), \(\left (0, -\frac {8}{3}\right )\)

conjugate axis length: \(\frac {2\sqrt {3}}{3}\)

[Picture]

\(r = \frac {2}{1 + \sin (\theta - \frac {\pi }{3})}\)

\(r = \frac {2}{1 + \sin (\theta - \frac {\pi }{3})}\) is the parabola \(r = \frac {2}{1 + \sin (\theta )}\) rotated through \(\phi = \frac {\pi }{3}\).

[Picture]

\(r = \frac {6}{3 - \cos \left (\theta + \frac {\pi }{4}\right )}\)

\(r = \frac {6}{3 - \cos \left (\theta + \frac {\pi }{4}\right )}\) is the ellipse \(r = \frac {6}{3 - \cos \left (\theta \right )} = \frac {2}{1 - \frac {1}{3} \cos \left (\theta \right )}\) rotated through \(\phi = -\frac {\pi }{4}\)

[Picture]

The matrix \(A(\theta ) = \left [ \begin{array}{rr} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{array} \right ]\) is called a rotation matrix.

We’ve seen this matrix most recently used in the proof of Theorem rotatecoordinatesthm.

Show the matrix from Example rotationmatrixex in Section MatArithmetic is none other than \(A\left (\frac {\pi }{4}\right )\).
Discuss with your classmates how to use \(A(\theta )\) to rotate points in the plane.
Using the even / odd identities for cosine and sine, show \(A(\theta )^{-1} = A(-\theta )\). Interpret this geometrically.