Lecture 3-The Ellipse The Hyperbola.pdf.pdf

Lecture 3
The Ellipse & The Hyperbola

The plane that intersects the cone is neither parallel
nor perpendicular to the axis of symmetry of the cone
and cuts through 2 “sides”.
Conic Sections - Ellipse

An ellipse is the locus of all points in a plane such that the sum of the distances
from two given points in the plane, “the foci” is constant.
Major Axis
Minor
Axis
Focus 1 Focus 2
Point
PF1 + PF2 = constant
The Ellipse

5
4
3
Example: Sketch the ellipse with equation 25x2 + 16y2 = 400
and find the vertices and foci.
1. Put the equation into standard form.
400
16
25 2
2
=
+ y
x divide by 400
25
16
1
400
16
400
25 2
2
2
2 y
x
y
x +
→
=
+ 1
=
2. Since the denominator of the y2-term
is larger, the major axis is vertical.
3. Vertices: (0, –5), (0, 5)
4. The minor axis is horizontal and intersects the ellipse at (–4, 0)
and (4, 0).
5. Foci: c2 = a2 – b2 → (5)2 – (4)2 = 9 → c = 3 foci: (0, –3), (0,3)
x
y
(4, 0)
(–4, 0)
(0, –3)
(0, 3)
(0, 5)
(0, –5)
So, a = 5 and b = 4.

The standard form of an ellipse centred at any point (h, k) with the major axis of
length 2a parallel to the x-axis and a minor axis of length 2b parallel to the y-axis,
is:
(x − h)2
a
2 +
(y − k)2
b
2 = 1
(h, k)
The Standard Forms of the Equation of the Ellipse

:Find the centre and the foci of the
Example
following equation then sketch it

Put in Standard Form
2 2
9 4 54 49 0
x y x y
+ − + + =
2 2
( 4 ) (9 54 ) 49
x x y y
− + + =
−
2 2
( 4 _) 9( 6 _) 49
x x y y
− + + + + =
−
2 2
( 4 4) 9( 6 9 9
) ( )
4 4 9
9
x x y y
− + + + + =
− + +
2 2
( 2) 9( 3) 36
x y
− + + =
2 2
( 2) 9( 3) 36
36 36 36
x y
− +
+ =
2 2
( 2) ( 3)
1
36 4
x y
− +
+ =
Group terms
Complete the square
Simplify each group
Divide by constant

Find the coordinates of the centre, the length of the major and minor axes, and
the coordinates of the foci of each ellipse:
F1(-c, 0) F2(c, 0)
b
c
a
a2 = b2 + c2
c2 = a2 - b2
Length of major axis: 2a
Length of minor axis: 2b
Vertices: (a, 0) and (-a, 0)
Foci: (-c, 0) and (c, 0)
Recall:
a
P
PF1 + PF2 = 2a
c
Analysis of the Ellipse

x2 + 4y2 - 2x + 8y - 11 = 0 (x − 1)2
16
+
( y +1)2
4
= 1
F1
F2
c = 2 3
(1 + 2 3, − 1)
(1- 2 3, - 1)
c = 2 3
Sketching the Graph of the Ellipse [cont’d]
Centre (1, -1)
(1, -1)

9x2 + 4y2 - 18x + 40y - 35 = 0 (x − 1)2
16
+
( y + 5)2
36
= 1
F1
F2
c = 2 5
c = 2 5
(1, − 5 + 2 5 )
(1
, -5 - 2 5)
Sketching the Graph of the Ellipse [cont’d]

The Hyperbola
The plane that intersects the cone is parallel to
the axis of symmetry of the cone.

The Hyperbola
Q
of the distances
difference
a set of points in a plane whose
–
Hyperbola
from two fixed points is a constant.
•

The Hyperbola
Transverse axis – the line that contains the foci and goes through
the center of the hyperbola.
Conjugate axis – the line that is
perpendicular to the transverse axis and
goes through the center of the hyperbola.
Conjugate axis
Center – the midpoint of the line
segment between the two foci.
Center
•

The Hyperbola
Identify the direction of opening, the coordinates of the center, the
vertices, and the foci. Find the equations of the asymptotes and
sketch the graph.
Vertices of transverse axis:
Equations of the Asymptotes
Foci
• •
•
•
• •

The Hyperbola
Vertices of transverse axis:
Foci
•
•
• •
•
•
Identify the direction of opening, the coordinates of the center, the
vertices, and the foci. Find the equations of the asymptotes and
sketch the graph.

The Hyperbola
Find b:
Center:
Equation of the Hyperbola
•
•
• •
•
•

The Hyperbola
Center:
Equations of the Asymptotes •
•
• •
•
•

The Hyperbola
Find the center, the vertices of the transverse axis, the foci and the
equations of the asymptotes using the following equation of a hyperbola.
Opening up/down

The Hyperbola
Vertices:
Foci:

The Hyperbola

The Ellipse
: Choose the correct answer
1
Example

Lecture 3-The Ellipse The Hyperbola.pdf.pdf

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