MTH-5105 Pretest A Solutions

1
MTH-5105 Pretest A Solutions

• Represent graphically
• x2 6x y2 2y 1 0

(x2 6x 9) (y2 2y 1) -1 9 1 (x
3)2 (y 1)2 9
C (3,1) r 3

1. Determine the equation of the circle below in
   general form.

C (-3,3) h -3 k 3 r 4
(x h)2 (y k)2 r2 (x (-3))2 (y 3)2
42 (x 3)2 (y 3)2 16 x2 6x 9 y2 6y
9 - 16 0 x2 y2 6x 6y 2 0
2

1. What is the equation of the tangent to the circle
   with equation (x 1)2 (y 2)2 5 with point
   of tangency (2,0)?

C (1,2) T(2,0)

1. Represent graphically and indicate the vertex,
   focus, axis of symmetry and directrix.

4a -8 a -2
h -½ k 3
Other Pts. Let x -2.5 (y 3)2 -8(-2.5
0.5) (y 3)2 -8(-2) (y 3)2 16 y 3 4
OR y 3 -4 y 4 OR y -1
Opens horizontally (left) Vertex (-0.5
,3) Focus (-2.5 ,3) Axis of Symmetry y
3 Directrix x 1.5
(-2.5,4) (-2.5,-1)
3

1. Determine the equation of the parabola below in
   its standard form.

h -3 k -1
a F V 0 (-1) 1
(x h)2 4a(y k) (x (-3))2 4(1)(y
(-1)) (x 3)2 4(y 1)

1. Represent the following inequality on a cartesian
   plane. Give the coordinates and indicate on the
   graph of the vertices and the foci and draw and
   state the asymptotes.

c2 a2 b2 c2 36 16 c2 52 c
7.2 F1(0, 7.2) F2(0, -7.2)
Asymptotes
4

1. Give the domain and range of this relation. Use
   either set-builder or interval notation.

Vertices (-1,-3) (-5,1) (-9,-3)
(-5,-7) Domain -9,-1 Range -7,1

1. Determine the expression for the 2 relations
   below. Express them in standard form.

a 4 b 6
a 5 b 3
5
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6

1. Find the equation of the parabola inscribed in a
   circle with centre (2,3) and whose radius is
   equal to 2. The abscissa of the vertex is 2. We
   know also that the parabola intersects the circle
   at abscissas 1 and 3.

Equation of parabola (x h)2 4a(y k) (x
2)2 4a(y 5) (3 2)2 4a(1.3 5) 1
4a(-3.7) -14.8a 1 a -0.068 (x 2)2
4(-0.068)(y 5) (x 2)2 -0.27(y 5)
Vertex (2,5)
Centre (2,3) Radius 2
Equation of circle (x h)2 (y k)2 r2 (x
2)2 (y 3)2 22 (x 2)2 (y 3)2 4
Let x 3
(3 2)2 (y 3)2 4 1 (y 3)2 4 (y 3)2
3 y 3 1.7 OR y 3 -1.7 y 4.7 OR y
1.3 (3,4.7) OR (3,1.3)
(3, 1.3) is a point on the circle and the
parabola.
7

1. The elliptical sign of cosmos restaurant
   measures 4 meters in length and 2 meters in
   width. The owner wants to attach the sign above
   the Os. The sign has a width of 1.6 meters at
   those points. How far apart will the attachments
   be separated.

Width 1.6 m
Separation 1.2 (-1.2) 2.4 m
8

1. Christian wants to hit his golf ball onto the
   green (area with the flag). This green is
   elevated 2 meters from the current position of
   his golf ball which lies 16 meters in front of a
   tree that is 9 meters high. When he hits the
   ball, it reaches a maximum height 1 meter
   directly over the tree. Knowing that the ball
   follows a parabolic trajectory, what horizontal
   distance does the ball travel when it strikes the
   green.

Equation of parabola (x 16)2 4a(y 10) (0
16)2 4a(0 10) 256 -40a a -6.4 (x 16)2
4(-6.4)(y 10) (x 16)2 -25.6(y 10)
Vertex (16,10)
Point (0,0)
Let y 2 (x 16)2 -25.6(2 10) (x 16)2
-25.6(-8) (x 16)2 204.8 x 16 14.3 OR x
16 -14.3 x 30.3 OR x 1.7 The longer
distance applies in this case. It strikes the
green after moving 30.3 meters.