Page Description Equation
710 standard equation of a circle with radius $a$ centered at the origin. \[ x^2+y^2=a^2 \]
710 standard equation of a circle with radius $a$ centered at the point $(h,k).$ \[ (x-h)^2+(y-k)^2=a^2 \]
711 standard form equations for parabolas \[ \]
714 standard equation of an ellipse centered at the origin with foci on the $x$-axis.
semimajor axis: $a$
semiminor axis: $b$
Center-to-focus distance: \[ c=\sqrt{a^2-b^2} \] Foci: \( (\pm\,c,0) \)
Vertices: \( (\pm\,a,0) \) 		\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] where $a\gt b.$
714 standard equation of an ellipse centered at the origin, with foci on the $y$-axis.
semimajor axis: $a$
semiminor axis: $b$
Center-to-focus distance: \[ c=\sqrt{a^2-b^2} \] Foci: \( (0,\pm\,c) \)
Vertices: \( (0,\pm\,a) \) 		\[ \frac{x^2}{b^2}+\frac{y^2}{a^2}=1 \] where $a\gt b.$
716 standard form equations for hyperbolas centered at the origin \[ \]
723 eccentricity \[ \]
724 eccentricity of hyperbola \[ \]
725 eccentricity of parabola \[ \]
725 focus-directrix equation \[ \]
728 quadratic curves, quadratic equations \[ \]
728 the cross product term \[ \]
729 equations for rotating coordinate axes \[ \]
729 Angle Rotation Formulas \[ \cot2\alpha=\frac{A-C}{B} \]
729 Angle Rotation Formulas \[ \tan2\alpha=\frac{B}{A-C} \]
731 Discriminant Test for the graph of the equation $Ax^2$ $+Bxy$ $+Cy^2$ $+Dx$ $+Ey$ $+F=0.$ \[ \begin{array}{cl} \textbf{Discriminant}&\textbf{Graph}\\ B^2-4AC=0&\text{parabola}\\ B^2-4AC\lt0&\text{ellipse}\\ B^2-4AC\gt0&\text{hyperbola} \end{array} \]

Notes

Page Notes
709 Definition. A circle is the set of points in a plane whose distance from a given fixed point in the plane is constant. The fixed point is the center of the circle; the constant distance is the radius.
711 Definition. A set that consists of all the points in a plane equidistant from a given fixed point and a given fixed line in the plane is a parabola. The fixed point is the focus of the parabola. The fixed line is the directrix.
712

Definition. An ellipse is the set of points in a plane whose distances from two fixed points in the plane have a constant sum. The two fixed points are the foci of the ellipse.

Fig 9.7 An ellipse. \( PF_1+PF_2 \) is constant.

Referring to fig 9.5, assume we drew this figure by fixing tacks at the arbitrary points $F_1$ and $F_2$ and winding a fixed-length string around them and a pencil at $P(x,y),$ so that the loop of string is depicted by the red lines in the figure. Since the length of the string, $F_1F_2+PF_1+PF_2,$ and the segment $F_1F_2,$ are both constant, then their difference, $PF_1+PF_2,$ is constant, too. However, $PF_1+PF_2$ is the sum of the distance from $P$ to two fixed points $F_1$ and $F_2.$ Therefore, by definition, the set of points $\{P(x,y)\}$ traced by the pencil is an ellipse, and $F_1$ and $F_2$ are its foci.

713

Definition. The line through the foci of an ellipse is its focal axis. The point on the axis halfway between the foci is its center. The points where the focal axis and ellipse cross are its vertices.

Fig 9.7 Ellipse defined by \( PF_1+PF_2=2a \) is the graph of the equation \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \)

713 Proof. Referring to fig 9.7, we shall prove the caption. The distance formula gives \[ PF_1 = \sqrt{(x+c)^2+y^2} \] \[ PF_2 = \sqrt{(x-c)^2+y^2} \] Furthermore, at $P(a,0),$ $PF_1=OF_1+OP=\abs{-c}+a=c+a$ and $PF_2=OF_2-OP=a-c$ so that $PF_1+PF_2$$=(c+a)+(a-c)$$=2a.$ Since, $PF_1+PF_2$ is constant for all $P,$ we obtain \begin{equation} \label{eq1} PF_1+PF_2=2a \end{equation} Combining the previous equations, we have \[ \sqrt{(x+c)^2+y^2} +\sqrt{(x-c)^2+y^2} =2a \] From this equation we obtain \begin{equation} \label{eq2} \frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1 \end{equation} by way of some lengthy algebra as follows: \[ \sqrt{(x+c)^2+y^2} =2a -\sqrt{(x-c)^2+y^2} \] \[ (x+c)^2+y^2 =\left(2a-\sqrt{(x-c)^2+y^2}\right)^2 \] \[ (x+c)^2+y^2 =4a^2-4a\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2 \] \[ \left[\frac{(x+c)^2-(x-c)^2-4a^2}{-4a}\right]^2 =(x-c)^2+y^2 \] \[ \left(\frac{4cx-4a^2}{-4a}\right)^2 =(x-c)^2+y^2 \] \[ a^2+\frac{c^2x^2}{a^2} =x^2+c^2+y^2 \] \[ (a^2-c^2)+\frac{c^2}{a^2}x^2-x^2 =y^2 \] \[ (a^2-c^2)+x^2\left(\frac{c^2}{a^2}-1\right) =y^2 \] \[ (a^2-c^2)+x^2\left(\frac{c^2-a^2}{a^2}\right) =y^2 \] \[ (a^2-c^2)-x^2\left(\frac{a^2-c^2}{a^2}\right) =y^2 \] \[ (a^2-c^2)\left(1-\frac{x^2}{a^2}\right) =y^2 \] \[ 1-\frac{x^2}{a^2} =\frac{y^2}{a^2-c^2} \] \[ \frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1\qquad \blacksquare \] To conclude the proof, we note that $b^2=a^2-c^2.$ To see that this is so, observe that $PF_1=PF_2$ at $P(0,b).$ Then, $2a=2PF_1$ at that point. Therefore, $a=PF_1,$ the hypotenuse of $\triangle abc.$ Since $a,b,c$ are constant for all $P,$ not just at $P(0,b),$ it follows that \begin{equation} \label{eq4} a^2=b^2+c^2 \end{equation} for all $P(x,y).$ Therefore, equation (\ref{eq2}) can be simplified to \begin{equation} \label{eq3} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \end{equation} and the proof is complete.
713 Definition. The major axis of the ellipse in equation (\ref{eq3}) is the line segment of length $2a$ joining the points $(\pm\,a,0).$ The minor axis is the line segment of length $2b$ joining the points $(0,\pm\,b).$ The number $a$ itself is the semimajor axis, the number $b$ the semiminor axis. The number $c,$ found from equation (\ref{eq4}) with \begin{equation}\label{eq5} c=\sqrt{a^2-b^2}, \end{equation} is the center-to-focus distance of the ellipse.
715 Definition. A hyperbola is the set of points in a plane whose distances from two fixed points in the plane have a constant difference. The two fixed points are the foci of the hyperbola.
715 Definition. The line through the foci of a hyperbola is the focal axis. The point on the axis halfway between the foci is the center of the hyperbola. The points where the focal axis and hyperbola cross are the vertices.
716 How to graph the hyperbola given by \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \]
  1. Mark the points $(\pm\,a,0)$ and $(0,\pm\,b)$ with line segments and complete the rectangle they determine.
  2. Sketch the asymptotes by extending the rectangle's diagonals.
  3. Use the rectangle and asymptotes to guide your drawing.
716 Definition. If an ellipse is revolved about its major axis to generate a surface, the surface is called an ellipsoid.
718 Definition. whispering gallery
730 Definition. possible graphs of quadratic equations