School of Mathematics and Statistics Te Kura Mātai Tatauranga

Slides

A pdf version of the slides, linking to here.

Latest version is here

From http://www.ics.uci.edu/~eppstein/junkyard/all.html I found this nice page (sadly a dead link now) http://britton.disted.camosun.bc.ca/jbconics.htm

Conics

Standard formulae and a diagram

Circle

Centered at $(0,0)$ with radius r.

\[x^2+y^2=r^2\]

\begin{tikzpicture} \draw[gray, very thin] (-3.1,-3.1) grid (3.1,3.1); \draw[red,<->] (-3.2, 0) -- (3.2,0); \draw[red,<->] (0,-3.2) -- (0,3.2); \draw[blue] plot[variable=\t,samples=1000,domain=0:360] ({cos(\t)},{sin(\t)}); \end{tikzpicture}

Ellipse

Centered at $(0,0)$, semi x-axis length is a, semi y-axis length is b.

\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]

\begin{tikzpicture} \draw[gray, very thin] (-3.1,-3.1) grid (3.1,3.1); \draw[red,<->] (-3.2, 0) -- (3.2,0); \draw[red,<->] (0,-3.2) -- (0,3.2); \draw[blue] plot[blue,variable=\t,samples=1000,domain=0:360] ({2*cos(\t)},{sin(\t)}); \end{tikzpicture}

Parabola

Centered at $(0,0)$, distance from the vertex to the focus is a.

\[y^2=4ax\]

\begin{tikzpicture} \draw[gray, very thin] (-3.1,-3.1) grid (3.1,3.1); \draw[red,<->] (-3.2, 0) -- (3.2,0); \draw[red,<->] (0,-3.2) -- (0,3.2); \draw[blue] plot[variable=\t,samples=1000,domain=0:1.4] ({(\t)^2},{2*\t}); \draw[blue] plot[variable=\t,samples=1000,domain=0:1.4] ({(\t)^2},{-2*\t}); \draw[dotted] (-1,3) -- (-1,-3); \end{tikzpicture}

Hyperbola

Centered at $(0,0)$, vertices are $(\pm a,0)$ and the asymptotes are $y=\pm \frac{b}{a}x$.

\[\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\]

\begin{tikzpicture} \draw[gray, very thin] (-3.1,-3.1) grid (3.1,3.1); \draw[red,<->] (-3.2, 0) -- (3.2,0); \draw[red,<->] (0,-3.2) -- (0,3.2); \draw[blue] plot[variable=\t,samples=1000,domain=-70:70] ({sec(\t)},{tan(\t)}); \draw[blue] plot[variable=\t,samples=1000,domain=-70:70] ({-sec(\t)},{tan(\t)}); \draw[dotted] (3,3) -- (-3,-3); \draw[dotted] (-3,3) -- (3,-3); \end{tikzpicture}

Foci properties

There are lots of these, we'll focus on the most straightforward ones, foci are on the real axis. ELI5.

Circle

Just the center and radius description. Easy peasy!

Ellipse

The pin-and-string description. Foci at $(-a,0)$ and $(a,0)$. Sum of the distances to the two foci is fixed.

Parabola

Distance equals width. Foci at a, vertical line through $-a$. The distance to the foci is equal to the horizontal distance to the line.

Hyperbola.

Like an ellipse, but difference instead of sum. The difference of the distances to the two foci is fixed.

Complex forms

Using the idea that $|z-(x+iy)|$ is the distance from z to $x+iy$.

Circle

Centered at $x_0+iy_0$ with radius r.

\[| z-(x_0+iy_0) | = r\]

Ellipse

Centered at $(0,0)$, foci at $-F$ and $F$, sum of distances is c.

\[ | z+F | + | z-F | = c\]

Parabola

Centered at $(0,0)$, foci at $(a,0)$. Trick is we want the real part of $z+a$, which is $\frac{z+a+\overline{z+a}}{2}$ which equals $\frac{z+\overline{z}}{2}+a$.

\[ | z - a | = \frac{z+\overline{z}}{2}+a\]

Hyperbola

Centered at $(0,0)$, foci at $(-a,0)$ and $(a,0)$, constant difference is d.

\[ | z+F | - | z-F | = d\]

Parametric forms

There are multiple possibilities, but here are the common ones.

Circle

\[x=r\cos t, y=r\sin t\]

Ellipse

\[x=a\cos t, y=b\sin t\]

Parabola

\[x=at^2, y=2at\]

Hyperbola

\[x=\frac{a}{\cos t}, y=a\tan t\]

Quadratic form

All conics can be written in this generic form, but I think that is above where we need to go.

\[Ax^2+Bxy+Cy^2+Dx+Ey+F=0\]

Tests on the coefficients determine what sort of conic it is.

Some problems to warm up in week 1

  1. What is the ellipse form of a circle centered at $(0,0)$ with radius r?
  2. Lets work out the slope of tangent lines of a circle.
    1. Make y the subject, making it a function by taking the positive square root, then differentiate. Which part of the circle is this?
    2. Same as before, but grab the other side of the circle.
    3. Now for a trick. Pretend y is shorthand for a function. This trick is called implicit differentiation. Now use the chain rule to show $y'=\frac{-x}{y}$.
    4. What is the slope of the lines perpendicular to those tangent lines?
    5. Sketch one on a circle graph. Where do those perpendicular (normal) lines always go through?
  3. It appears that the hyperbola tries to reach the asymptotes.
    1. Rewrite the hyperbola form in the form $y^2=K+Cx^2$, clearly identify C and K in terms of a and b.
    2. As $x$ gets large, the K term gets swamped, so we have $y^2\approx Cx^2$. Assume they are actually equal, and solve to get the two asymptotes.
  4. Sliding shapes is pretty easy, moving $(0,0)$ to $(x_0,y_0)$ only requires replacing x with $x-x_0$ and y with $y-y_0$
    1. Write the forms for each conic (ellipse, parabola and hyperbola) with the new center $(x_0,y_0)$.
    2. Use sliding to explain why $y=(x-x_0)^2+y_0$ is the $y=x^2$ graph moved so the turning point is $(x_0,y_0)$, and use calculus to check this is true.
  5. These are all conics, use algebra to identify which type it is and the relevant details.
    1. $x^2+y^2+2x=0$ (not centered at $(0,0)$)
    2. $4x^2+9y^2=36$
    3. $(x-y)(x+y)=4$
    4. $(x-y)(x+y)=0$ this isn't a proper conic, but it is familiar and is technically a degenerate conic.

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