# Further Maths - Conic Sections

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/conic-sections/ · Updated: 2021-10-05 · Tags: school, further-maths, further-pure-1, conic-sections, hyperbola

## 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.

![PHOTO RECTANGULAR HYPERBOLA](rectangular-hyperbola.png)

##### 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??
![PHOTO RECTANGULAR HYPERBOLA GRAPH](rectangular-hyperbola-graph.png)

##### 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 $y^2 = 4ax$??
1. Implicit differentiation and rearranging
2. 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 $y^2$??
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
##### What is the "foot" of perpendicular from the origin to a line??
The point where you'd draw the little 90 degree square.

### 2022-05-29
##### Is it better to use things like $\tanh$ or $\cot$, or things like $\frac{\sinh}{\cosh}$ or $\frac{\cos}{\sin}$ in conic sections questions??
The latter, keep it expanded

##### $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ 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 $(a, 0)$??
$$
x = a
$$

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