Embed Size (px)
Citation preview
The EllipseThe Ellipse
€
a + b = constant
QuickTime™ and aGraphics decompressor
are needed to see this picture.
ab
€
x 2
a2 + y 2
b2 =1
b
a
€
x 2
16+ y 2
9=1
3
4
When the size of a becomes the
same as b, we get a circle.
Definition: is the locus of all points which meet the
condition... sum of distances to each
focus is constant
QuickTime™ and aGraphics decompressor
are needed to see this picture.
c is the focusRelation to the focus:
€
a2 = c 2 + b2
a
b
c
a
€
a2 = b2 + c 2
QuickTime™ and aGraphics decompressor
are needed to see this picture.
Sketch the following graphsSketch the following graphs
€
A. x −1( )
2
25+
y + 3( )2
16=1
B. 4x 2 + y 2 = 4
C. x 2 + 4y 2 − 2x +16y +15 = 0
-4
-2
0
2
4
6
8
10
12
14
16
-2 -1 0 1 2 3 4 5 6€
x − 2( )2
16+
y − 6( )2
64=1
Write the equation of this ellipse.
-6
-4
-2
0
2
4
6
8
10
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
Write the equation of this ellipse.
€
x + 4( )2
9+
y − 4( )2
16=1
Parametric equations of the ellipse
€
x = acosθ, and y = bsinθ
Eliminate to obtain the equation of the ellipse
€
θ
Parameter
Write down the parametric Write down the parametric equations for these ellipsesequations for these ellipses
€
A. x 2
36+ y 2
9=1
B. x 2 + 4y 2 =1
C. 4x 2 + y 2 + 8x +10y +13 = 0
Find the equation of the tangent to the ellipse at the point where
€
θ =π3
€
4x 2 + y 2 =16
A line tangent to an ellipse at a point P
makes equal angles with the lines through P and
the foci.
It follows that a ray of light It follows that a ray of light emanating from one focus of emanating from one focus of
an ellipse will be reflected an ellipse will be reflected through the other focus.through the other focus.
QuickTime™ and aGIF decompressor
are needed to see this picture.
QuickTime™ and aGraphics decompressor
are needed to see this picture.
QuickTime™ and aGraphics decompressor
are needed to see this picture.
QuickTime™ and aGraphics decompressor
are needed to see this picture.
A point moves so that the sum of its distances from the two
points (1, 1) and (3, 1) is always 10. Find the equation of the
locus of the point.
HomeworkHomework Exercises 13.5, 13.6Exercises 13.5, 13.6 Delta Exercise 14.1Delta Exercise 14.1 Number 7 onwardsNumber 7 onwards
