MAT01B1: Parabolas and Hyperbolas · Parabolas: a parabola is the set of points in a plane that are...
Transcript of MAT01B1: Parabolas and Hyperbolas · Parabolas: a parabola is the set of points in a plane that are...
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MAT01B1: Parabolas and Hyperbolas
Dr Craig
30 October 2018
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My details:
I Consulting hours:
Monday 14h40 – 15h25
Thursday 11h20 – 12h55
Friday 11h20 – 12h55
I Office C-Ring 508
https://andrewcraigmaths.wordpress.com/
(Or, just google ‘Andrew Craig maths’.)
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Sick Test
I Today
I 15h30 – 17h00
I D1 LAB 308
I Don’t be late
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Parabolas: a parabola is the set of points
in a plane that are equidistant from a fixed
point F and a fixed line. The point F is
called the focus and the line is called the
directrix.
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Parabolas
The vertex is the point of the parabola that
is on the line perpendicular to the directrix
that goes through F .
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A parabola with focus (0, p) and directrix
y = −p has equation
x2 = 4py
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If we interchange x and y we get
y2 = 4px, focus (p, 0), directrix x = −p
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Parabola examples
Find the focus and directrix of the parabola
y2 + 10x = 0.
Find the vertex, focus and directrix of the
the parabola 2x = −y2.
A shifted parabola:
Sketch y2 + 2y − x = 0.
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Hyperbolas
A hyperbola is the set of all points in a
plane the difference of whose distances from
two fixed points F1 and F2 is a constant.
|PF1| − |PF2| = ±2a
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We can show that if the foci of a hyperbola
are on the x-axis at (±c, 0) and we have
|PF1| − |PF2| = ±2a, then the equation is
x2
a2− y2
b2= 1 (c2 = a2 + b2)
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Equations of a hyperbola (1)
The hyperbola
x2
a2− y2
b2= 1
has foci (±c, 0), where c2 = a2 + b2,
vertices (±a, 0), and asymptotes
y = ±(b/a)x.
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Equations of a hyperbola (2)
We can also have the foci of a hyperbola on
the y-axis.
The hyperbola
y2
a2− x2
b2= 1
has foci (0,±c), where c2 = a2 + b2,
vertices (0,±a), and asymptotes
y = ±(a/b)x.
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Important hyperbolic facts
I a hyperbola where the x2 has a positive
coefficient looks like an x
I the value of a is used to find the
coordinates of the vertices
I the a2 is always below the term with the
positive coefficient
I there is no required relationship between
a and b
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Hyperbola example 1
Find the foci and asymptotes of the
hyperbola 9x2 − 16y2 = 144 and sketch its
graph.
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Hyperbola example 2
Find the foci and equation of the hyperbola
with vertices (0,±1) and asymptote y = 2x.
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Sketch of example 2: y2 − 4x2 = 1
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Yet another shifted conic:
Consider the curve
4x2 − y2 − 24x− 4y + 16 = 0.
We can tell immediately that this is a
hyperbola. However we must first do some
algebra before deciding which shape it will
be.
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4x2 − y2 − 24x− 4y + 16 = 0
(x− 3)2
4− (y + 2)2
16= 1
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Shifted example 2
Sketch and find the foci of the conic
9x2 − 4y2 − 72x + 8y + 176 = 0.
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Online sketcher
Use the following website to help familiarise
yourself with conic sections:
https://www.desmos.com/calculator/vgfqbejegx
To change it to the equation of an ellipse
just click the function box and change the
minus to a plus.
You can also click the 3 horizontal lines in
the top left corner to sketch other types of
curves.
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A shifted ellipse: consider
(x− 2)2
3+
(y + 1)2
2= 1.
This will be the same shape as the ellipse
x2
3+
y2
2= 1
but shifted 2 units to the right and 1 unit
down. Also, we have a =√3 and b =
√2.
Now, find the centre, vertices and foci and
then sketch 4x2 + y2 − 8x + 4y + 4 = 0.
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(x− 2)2
3+
(y + 1)2
2= 1
(x− 1)2 +(y + 2)2
4= 1