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[j]IRIIW INNOVATION!
-a.u Engineering May 2019
THE CONIC SECTIONS:
Analytic Geometry 2
THE PARABOLA (e = 1):
THE ELLIPSE (e < 1):
= set of points which are equidistant from the focus
= set of points such that the sum of the distances
from any point to the two fixed points (foci) is
as well as from the directrix.
constant. The constant sum is equa l to the length of
the major axis, 2a.
y
X
~ ---Latus rectum
2a
a
Id1 + d2=d3 + d4 =ds + d5I
Ellipse
General Definition of Conic Sections:
= a set of points such that the distance from any point to
a fixed point called focus is in constant ratio to its distance
to a fixed line called directrix. The constant ratio is called
its eccentricity, e.
y
(x - a)2 + y 2 = x + a
x
2
·- 2ax + a
2
+y
y
V
2
= x 2 + 2ax + a 2
2
= 4ax
V
e = fJd1 = fz/d2 = f3/d3
If e = 1, it's a parabola
Standard Equation of the Ellipse:
f1 = d1
D
x2 = -4ay
C
Let constant sum = 2a. Then
x2 = 4ay
d 1 +d 2 =2a
y
e < 1, it's an eJlipse
✓(x+c)2 + y 2 + ✓(x - c)2 +y 2 =2a
e > 1, it's a hyperbola
2
y = - 4ax
2
y = 4ax
Directrix
Note: If vertex is at (h, k), change x to x-h and y to y-k.
Length of latus rectum = 4a
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Simplifying,
X
2
y
2
z+
2
2 =1
a
a -c
2
2
Let b = a2 - c
2
X
y
2
-. + -=1
b2
a2
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[j]IIIYIIW INNOYITION1
CJwi1 Engineering May 2019
y
X
X
2
y
2
2+2= 1
a
b
Problems: (PARABOLA)
1. Locate the vertex and focus and find the leng~h of
the latus rectum of the parabola:
x2 + 12x + l0y + 6 = 0
9. A concrete arch bridge is to be built in the form
of a semi-ellipse. It must span a width of 6m. In
addition, the central 4.2m of it must be a t least 2.4m
high. How high is the arch at midspan?
2. A parabola has its axis parallel to the y-axis, one
end of its latus rectum is at (9, 6) and the vertex is
at (5, 4). Determine the following:
a) length of latus rectum
b) equation of the p~rabola
c) equation of the directrix of the parabola.
10. The distance (center to center) of the moon from
the earth varies from a minimum of 221,463 miles
to a maximum of 252,710 miles. Find the
eccentricity of the moon'. s orbit.
3: Find the equation of the locus of a point which
moves so that its distance from the line x + 4 = 0 is
5 more than its distance from the point (3, 1).
Note: If center is at C(h, k), just change x to (x-h)
and y to (y-k).
Properties :
01. a
2
= b 2 + c2
a>c ;
a>b
2b
2
02. length of latus rectum= - a
4. A comet from deep space approaches the sun
along a parabolic orbit. When the comet is 100
million miles from the sun, the line joining the sun
and the comet makes an angle of 60° with the axis
of the parabola. How close to the sun will the comet
get?
5. A parabolic concrete arch spans a width of 40ft
with a 20ft wide road passing under the bridge. The
minimum vertical clearance over the roadway must
be 10ft. What is the height of the smallest such arch
that can be used?
C
03. e = - < 1
a
Problems: (ELLIPSE)
6. Locate the center and foci and determine the
length of the latus rectum of the ellipse whose
equation is:
16x2 + 25y2 - 128x - 150y + 381 = 0
a
04. d = e
. 05. Area, A
= rrab
.
✓ a2+b2
06. Perimeter, P = 2rr
_
2
C
07. 2nd eccentricty, e = -
.
b
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Analytic Geometry 2:
,. 7. Find the equation of the ellipse that has its center
at (1, 1), a vertex at (3, 1), and that passes through
the ori~in.
Problem for Practice:
1. Locate the vertex, focus and the coordinates of
the length of the latus rectum of the parabola:
y 2 + 4x - 4y + 16 = 0
Answer: V(-3, 2); F(-4, 2);
Ends of Latus Rectum (-4, 4) and (-4, 0)
I
2. Find the equation of the parabola whose vertex is
the origin and whose directrix is the line x = 4.
Answer: y 2 = -16x
3. An arch in the shape of an arc of a parabola
measures 6m across the base and its vertex is 2.50m
above the base. Find the length of the beam parallel
to the base and 2m above it.
Answer: 2.68 m
4. Find the equation of the ellipse whose vertices are
the points (4, 6) and (4, -2) and whose eccentricity is
3/4.
(x-4) 2
(y-2) 2
Answer-• -7- + -16-
=1
5. A cross-section of a trough is a semi-ellipse with
width at the top 18cm and depth 12cm. The trough
is filled with water to a depth of 8cm. Find the
width at the surface of the water.
Answer: 12.../2 cm
8. Find the eccentricity of an ellipse whose major
axis is twice as long as its minor axis.
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J. f'nt the equation of the locus of a point which moves so that its distance
the line x + 4 =0 is 5 more than its distance from the point (3, 1).
•
X + 4 =✓ (X • 3f + (y •
W+ 5
y
x+4=0
or x =-4
6. locate the center and foci and determine the Jength of the
latus rectum of the ellipse whose equation is:
16x2 + 25y2 - 128x -150y + 381 = O .
Solution:
I
x-1 = ✓ (x-3f+(y-1f
I
(x-1)2= (x-3f+(y~1-f
I
I
I
I
/ d2
16x2 -128x
+25y2 -150y
=-381
16 (x2 - 8x + 16 ) + 25 (y2 - 6y + 9) = -381 + 256 + 225
1
1
,
I
I
25 (y- 3) 2
+
16(x-4)2
'
(x-4)2
25/4
4x-8=(y - 1f
(y-3)2·
+
4
l(y-1)2 = 4 (x-2) I *LOCUS OF APOINT
Parabola that opens to the right
a= 5/2
b=2
y
:
y
,-<
>t(
I
I
>'1
--~
-
=1
a2 ..----- b2
(4, 3)
1c
:
: C : C :
= 100
; A(3, 1)
· x2 - 2x + 1 = x2 • 6x + 9 + (y • 1f
:
1
F' (2.5, 3)
a2 = b2 + c2
(5/2)2 = (2)2 + c2
c=3/2
LR= 2b2 / a
_= 2(2)2 / (5/2)
F (5.5, 3)
ILR = 3.2 units I
ffi)
maximum
minimum
distance
distance
V
V'
Y ~ a x i s of_parabola
f 1 "' 100 x 10s miles
..
\ ...
·
100 x 108 ·c os 60°
j ,,
, ........
252,710
221,463
C
a
a
C
e=-a
2a = 221,463 + 252,710
= 237,086.5 · 221,463
c = 15,623.5 miles
a = 237,086.5 miles
e = is.623 .s
closest position
to the sun
1
C
I= o. 066 I
237,086 .5 .___ ____,
,
a
for p~rabola, e = 1
directrix ...--/'
100 X 106 = 2a + 100 X 106 COS 60°
Ia= 25 x 10 milesj
6