Flashcards
Parabolas
How can you form a parabola from a cone??
Slice it parallel to its slope.
Why must you slice a cone PARALLEL to the slope to form a parabola??
Otherwise you’d either get an ellipse or intersect the cone twice and get a hyperbola.
What is the parametric equation that defines a parabola??
$$ x = at^2 $$ $$ y = 2at $$
When thinking about parabolas as a conic section, is it better to think of them symmetrical around the $x$-axis or $y$-axis??
$x$-axis.
What is Cartesian definition of a parabola??
$$ y^2 = 4ax $$
What is the focus-directrix definition of a parabola??
The locus of points that are the same distance from a fixed focus to a fixed straight line called the directrix.
What is the focus of a parabola??
The point that the locus must be the same distance to from the directrix.
What is the directrix of a parabola??
The line that the locus must be the same distance to from the focus.
What are the co-ordinates of the focus for a parabola $y^2 = 4ax$??
$$ (a, 0) $$
What is the equation of the directrix for a parabola $y^2 = 4ax$??
$$ x + a = 0 $$
What is the vertex of a parabola??
Its turning point.
What is the axis of a parabola??
Its line of reflectional symmetry.
What are the co-ordinates of the vertex for a parabola $y^2 = 4ax$??
$$ (0, 0) $$
What is the Cartesian equation of the parabola with focus $(7, 0)$ and directrix $x + 7 = 0$??
$$ y^2 = 28x $$
If the focus of a parabola is $(5, 0)$, what is the equation of the directrix??
$$ x + 5 = 0 $$
2021-12-01
Rectangular Hyperbola
How can you form a rectangular hyperbola??
Slice the cone perpendicular to its base so that it intersects both halves.
How can you form a hyperbola from a cone??
Slice the cone so that you intersect both halves.
What are the two sections of a hyperbola called??
Branches.
What does the graph of a rectangular hyperbola look like on a pair of axes??
What is the nice, implicit equation for a rectangular hyperbola??
$$ xy = c^2 $$
What is the parametric equation for a rectangular hyperbola??
$$ x = ct $$ $$ y = \frac{c}{t} $$
What is special about a rectangular hyperbola compared to a normal hyperbola??
The asymptotes are perpendicular to eachother (consider how the axes meet).
Where are the two asymptotes for a rectangular hyperbola??
At $x = 0$ and $y = 0$.
What type of curve is $xy = 64$??
A hyperbola.
What is $c$ for $xy = 8$??
$$ c = 2\sqrt{2} $$
2022-01-25
What two techniques could you use to work out the gradient at a point on a parabola $y^2 = 4ax$??
- Implicit differentiation and rearranging
- Parametric differentiation
2022-01-31
How can you find the slope of the tangent to a rectangular hyperbola $(ct, c/t)$??
Use parametric differentiation.
How can you prove that a parabola is the locus of points an equal distance away from a focus and a directrix??
Set up a statement saying that the distances are equal and rearrange for $y^2 = 4ax$.
2022-02-02
If the line with equation $y = mx + c$ is a tangent to the parabola with equation $y^2 = 4ax$, how could you show $a = mc$??
Set the $y$s equal to each other and use the fact the discriminant must be equal to $0$.
What is the general Cartesian equation for an ellipse??
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
In order to work out $a$ and $b$, what must every Cartesian ellipse equation be equal to??
$$ 1 $$
If $$4x^2 + 9y^2 = 36$$ how could you work out the values of $a$ and $b$ for the ellipse??
Divide both sides by $36$.
What is the parametric equation for an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$??
$$ (a\cos t, b\sin t) $$
What is the general Cartesian equation for an hyperbola??
$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$
What are the two possible parametric equations for a hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1$$??
$$ (\pm a \cosh t, b \sinh t) $$ $$ (a \sec t, b \tan t) $$
What is the advantage of using the $(\pm a\cosh t, b\sinh t)$ parametric equations over $(a \sec t, b \tan t)$a??
You don’t need to specify a domain for $t$.
What is the domain for $t$ in the parametric equations for a hyperbola $(a \sec t, b \tan t)$??
$$ -\pi \le t < \pi, \t \ne \pm \frac{\pi}{2} $$
Where are the asymptotes of a hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$
$$ y = \pm \frac{b}{a} x $$
Why aren’t most hyperbolas “rectangular” hyperbolas??
Because their asymptotes aren’t perpendicular to one another.
2022-02-04
What is the eccentricity of a conic section??
The ratio of the distance to the focus vs the distance to the directrix.
If $$\frac{\text{distance to focus}}{\text{distance to directrix}} = e$$ how can you work out the distance to the focus given the distance to the directrix??
$$ \text{distance to focus} = e \times \text{distance to directrix} $$
If $e = 1$ then what conic section do you get??
A parabola.
If $e < 1$ then what conic section do you get??
An ellipse.
If $e > 1$ then what conic section do you get??
A hyperbola.
What’s the general strategy for showing that a certain conic section has Cartesian equation given the locations of the foci and directrices??
Show that the squared distances are equal to a ratio.
When working out the eccentricity of an ellipse, what do you need to consider??
Whether $a > b$ or vice versa.
Why is it important whether $a > b$ or $b > a$ when working out the foci and directrices of an ellipse??
Because it’s like the ellipse has been rotated, so the foci and directrices need to be rotated too.
When $a > b$ what are the coordinates for the foci of an ellipse in terms of $a$ and $e$??
$$ (\pm ae, 0) $$
When $b > a$ what are the coordinates for the foci of an ellipse in terms of $b$ and $e$??
$$ (0, \pm be) $$
When $a > b$ what are the equations for the directrix of an ellipse in terms of $a$ and $e$??
$$ x = \pm \frac{a}{e} $$
When $b > a$ what are the equations for the directrix of an ellipse in terms of $b$ and $e$??
$$ y = \pm \frac{b}{e} $$
If you’ve got to this stage $$x^2(1 - e^2) + y^2 = a^2(1 - e^2)$$ when proving the Cartesian equation of a hyperbola, how can you flip it so you have a minus sign in front of the $y^2$??
Multiple the $(1 - e^2)$
$$ x^2(e^2 - 1) - y^2 = a^2(e^2 - 1) $$
2022-02-08
How would you show that a certain parametric equation satisfies some sort of $$f(x, y) = g(x, y)$$??
Substitute in the parametric equation for both sides and verify that they’re equal.
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Metadata
date: 2021-10-05 16:32
tags:
- '@?public'
- '@?school'
- '@?further-maths'
- '@?further-pure-1'
- '@?conic-sections'
- '@?hyperbola'
title: Further Maths - Conic Sections
