Further Maths - Conic Sections
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?
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?
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 $y^2 = 4ax$?
What is the equation of the directrix for a parabola $y^2 = 4ax$?
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 $y^2 = 4ax$?
What is the Cartesian equation of the parabola with focus $(7, 0)$ and directrix $x + 7 = 0$?
If the focus of a parabola is $(5, 0)$, what is the equation of the directrix?
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?
What is the parametric equation for a rectangular hyperbola?
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 $xy = 8$?
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?
In order to work out $a$ and $b$, what must every Cartesian ellipse equation be equal to?
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
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]??
\[(a\cos t, b\sin t)\]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 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 parametric equation for an ellipse
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
?
What is the general Cartesian equation for an hyperbola?
Where are the asymptotes of a hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] \[y = \pm \frac{b}{a} x\]What are the two possible parametric equations for a hyperbola
\[\frac{x^2}{a^2} - \frac{y^2}{a^2} = 1\]
?
2022-02-04
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 $t$.
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??
\[\text{distance to focus} = e \times \text{distance to directrix}\]What is the domain for $t$ in the parametric equations for a hyperbola $(a \sec t, b \tan t)$?
Why aren’t most hyperbolas “rectangular” hyperbolas?
Because their asymptotes aren’t perpendicular to one another.
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?
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.
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 $y^2$?? Multiply 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.
2022-03-29
When $a > b$ what are the coordinates for the foci of an ellipse in terms of $a$ and $e$?
2022-05-29
When $b > a$ what are the coordinates for the foci of an ellipse in terms of $b$ and $e$?
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\[\frac{x^2}{16} - \frac{y^2}{9} = 1\]What is the equation of the tangent to this hyperbola at $(4, 0)$??
\[x = 4\]#####
\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]What is the equation of the tangent to this hyperbola at $(a, 0)$??
\[x = a\]