Conics - Mags Maths

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Conics
Merit - Excellence
Question 1
A cross-section of a parabolic reflector is shown
in the figure. A bulb is located at the focus.
• (a) Find an equation for the parabola in
parametric form relative to the axes shown.
• (b) Find the position of the bulb relative to the
axes shown.
• General form of the
equation:
y = kx
2
Substitute for x and y
y = kx
2
20 = 16k
k = 25
2
y = 25x
2
y = 4 ( 6.25 ) x
2
Þ a = 6.25
Parametric form of the equation
a = 6.25
x = at , y = 2at
2
x = 6.25t , y = 12.5t
2
Position of the bulb (6.25, 0)
Question 2
The greatest and least distances from a seat on
an elliptical ride at a fair ground to a focal point
of the ellipse are 18 m and 2 m respectively.
• (a) Find the equation of the locus of the seat
relative to the axes shown.
• (b) How high is the seat from the ground
when the seat is in the position shown?
Put the information on the diagram
2m
18m
c = 8, a = 10
2m
18m
c = 8, a = 10
2m
18m
a =b +c
2
2
Þb=6
2
Equation is
2m
y - 8)
x
(
+
=1
2
2
10
6
2
18m
2
Substitute x = 5
2m
18m
y - 8)
5
(
+
=1
2
2
10
6
y = 8 + 3 3 = 13.2m
2
2
Question 3
A ship at position S receives radio signals from two
radio stations located at A and B which are 640 km
apart. The ship’s radio operator observes that the
signals he receives are 1200 microseconds apart.
Assuming that radio signals travel at a speed of 300
m/microsecond.
• (a) Choosing a suitable coordinate system, write
down an equation that describes the path of the
ship.
• (b) If the ship is due north of B, how far off the
coastline is the ship?
We are dealing with a hyperbola and
will assume distance A – B is constant
It could be two possibilities but the picture indicates
that distance to A is greater
c = 320
d1 - d2 = 1200 ´ 300m
= 360km
= 2a
a = 180
The situation tells us that the locus is a hyperbola
c = 320
a = 180
c2 = a2 + b2
b = 70000
x2
y2
=1
2
180 70000
If the ship is due north of B, how far off the coastline is
the ship? (Assume A-B are West-East)
x2
y2
=1
2
180 70000
x = c = 320
320
y
=1
2
180 70000
y = 389km
2
2
Question 4
• Find the equation of the tangent to the curve
4x - 9y = 6
2
• at the point (2, 1).
2
X = 2, y = 1
4x - 9y = 6
2
2
dy
8x -18y = 0
dx
dy 4 x 8
=
=
dx 9y 9
y = mx + c
8
1= ´2+c
9
7
c=9
4
7
y= x9
9
Question 5
• John shoots apples from his ‘potato canon’ over
the edge of a cliff as shown on the diagram. He
stands 6 m from the cliff edge and his apples
reach a height of 4 m as shown. The path of his
apples forms a parabola.
• (a) Find an equation for the parabola relative to
the axes shown.
• (b) Find the height, h, of the cliff above the river
bank if his apple lands on the edge of the river
bank which is 15 m from the face of the cliff.
y = kx 2 + 4
x = -6, y = 0
0 = 36k + 4
1
k =9
1 2
y=- x +4
9
(b) Find the height, h, of the cliff above the river bank if
his apple lands on the edge of the river bank which is
15 m from the face of the cliff.
x = 15
1 2
y=- x +4
9
1
y = - ´15 2 + 4
9
y = -21
h = 21
Question 6
• A tank, as shown, for storing petrol has an
elliptical cross-section. The tank is 6 m wide
and 4 m high.
• (a) Find the equation of the cross-section
relative to the axes shown.
• (b) Find the equation of the cross-section (in
parametric form) relative to the axes shown.
• (c) Find the depth, d, of the tank 0.8 m from
the edge.
Find the equation of the cross-section
relative to the axes shown.
x y
+ 2 =1
2
3 2
2
2
Find the equation of the cross-section (in
parametric form) relative to the axes shown
x y
+ 2 =1
2
3 2
x = 3cosq
2
2
y = 2 sin q
Find the depth, d, of the tank 0.8 m from the
edge.
x y
+ 2 =1
2
3 2
x = 2.2
2
2
y = 1.36m
d = 1.36 + 2 = 3.36m
Question 7
• An spy plane flies between two towns A and B
which are 420 km apart, so that it is always 60
km closer to B than to A.
• (a) Choosing a suitable coordinate system,
write down an equation that describes the
path of the spy plane.
• (b) If the plane is due south of B, how far is
the plane from town A?
Draw a diagram
d1
d2
Hyperbola “always 60 km closer to B
than to A”
d1
d2
2a = 60
a = 30
c = 210
c2 = a2 + b2
b = 43200
2
x
y
=1
900 43200
2
2
If the plane is due south of B, how far is the plane from
town A? Assume AB are West-East
d1
x = c = 210
d2
210
y
=1
900 43200
y = -1440km
2
2
If the plane is due south of B, how far is the plane from
town A? Assume AB are West-East
d1
d2
210
y
=1
900 43200
y = -1440km
2
d A = 420 2 +1440 2
= 1500km
2
Question 8
• A parabolic reflector has measurements as
shown on the diagram.
• (a) Find an equation for the parabola relative
to the axes shown.
• (b) If a light source is placed at the focus point,
the reflector will produce a beam of parallel
rays. Find the position of a light source if its
beam is to produce parallel rays.
y = k ( x -100 )
2
400 2 = k ( 500 -100 )
k = 400
y = 400 ( x -100 )
2
Find the position of a light source if its
beam is to produce parallel rays.
The light source is 100mm
from the vertex.
y 2 = 400 ( x -100 )
4a = 400
a = 100
Question 9
• A lake is in the shape of an ellipse. The lake is
2 km wide and 10 km long. George’s house is
situated 1.5 km from the eastern end of the
lake as shown.
• (a) Find the equation of the cross-section (in
parametric form) relative to the axes shown.
• (b) If he rows across the lake to Tom’s house
and back every morning, how far does he row
every morning in total.
x = 5 cosq
y = sin q
If he rows across the lake to Tom’s house and back every
morning, how far does he row every morning in total.
x 2 y2
+ =1
25 1
x = 3.5
y = 0.714
distance = 2.86 km
Question 10
• Suppose a gun is heard first by one observer and then
1.5 seconds later by a second observer in another
location a distance of 3 km from the first observer.
Assume that the gun and the ear levels of the
observers are all in the same horizontal plane and that
the speed of sound at the time was 340 m/s.
• (a) Choosing a suitable coordinate system, write down
an equation that describes the possible position of the
gun.
• (b) If the gun is due north of the first observer, how far
is the gun from the second observer?
d2 - d1 = 1.5 ´ 340
1.5 = 0.255 + b
2
= 0.51km
b 2 = 2.184975
2a = 51
a = 0.255
c = 1.5
2
d2
d1
2
1.5 = 0.255 + b
2
a = 0.255
x
y
=1
2
0.255 2.184975 d2
2
2
2
b 2 = 2.184975
d1
2
If the gun is due north of the first observer, how far is
the gun from the second observer?
x
y
=1
2
0.255 2.184975
d2
x = 1.5
2
2
y = 8.57
d = 32 + 8.57 2 = 9.1km
d1
Question 11
• A cross-section of a parabolic reflector is
shown in the figure. The bulb is located at the
focus and the opening at the focus is 10 cm.
• (a) Find an equation for the parabola relative
to the axes shown.
• (b) Find the diameter of the opening (CD), 12
cm from the vertex.
Latus Rectum is 10
y =10x
2
Find the diameter of the opening (CD),
12 cm from the vertex.
y =10x
2
x =12
y =10.95
Diameter = 21.9cm
Question 12
• A semi-elliptic arch spans a motorway 50 m
wide.
• (a) Find an equation for the ellipse in
parametric form relative to the axes shown.
x 2 y2
+ 2 =1
2
25 b
x = 15, y = 14
2
2
15 14
+ 2 =1
2
25
b
b 2 = 306.25
x
y
+
=1
2
25 306.25
2
2
(b) How high is the arch if a centre section of the
highway 30 m wide has a minimum clearance of
14 metres.
x
y
+
= 1 Þ b = 17.5m
2
25 306.25
2
2
Question 13
An aircraft flies between two radio beacons A
and B, which are 100 km apart, so that it is
always 40 m closer to A than to B.
• (a) Choosing a suitable coordinate system,
write down an equation that describes the
path of the aircraft.
• (b) What approximate path would the aircraft
be flying when it is 1000 km from A?
c = 50
2a = d1 - d2 = 0.04
a = 0.02
c2 = a2 + b2
d1
d2
A
100 km
b = 2500
2
B
x2
y2
=1
2
0.02 2500
d1
d2
A
100 km
B
What approximate path would the aircraft be flying
when it is 1000 km from A?
d1
d2
A
100 km
x2
y2
=1
2
0.02 2500
50
y=±
x
0.02
y = ±2500x
B
Question 14
Find the equation of the tangent to the curve
y2 - 2y - 4x + 4 = 0
• at the point (1, 2)
Question 14
y 2 - 2y - 4x + 4 = 0
dy
-4 + ( 2y - 2 ) = 0
dx
dy
2
2
=
=
=2
dx y -1 2 -1
Question 14
y 2 - 2y - 4x + 4 = 0
dy
-4 + ( 2y - 2 ) = 0
dx
dy
2
2
=
=
=2
dx y -1 2 -1
x = 1, y = 2
2 = 2+c
y = 2x

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