Review for test 4

Review for test 4
1) Solve the system of equations.
 xy  8
 x  y  6
a) 
 2
b)  x  xy  20
x  2 y  3
2) Solve the system of equations by substitution.
2 x  3 y  4

 x  5 y  6
3) Graph the equation.
( x  2) 2 ( y  1) 2

1
4
9
4) Find the center, vertices, foci, transverse axis and the equations of asymptotes of given hyperbolas
a) x2 – 4y2 - 2x - 24y - 39 = 0
9y2 - 16 x2 = 144
2
2
c) ( x  2)  ( y  3)  1
4
9
b)
1
2
5) Find the equation, in the standard form, of the hyperbola with vertices (0,± 4) and asymptotes y   x
6) Find the equation, in the standard form, of the hyperbola with vertices (± 5,0) and foci (±8, 0)
7) Find the equation, in the standard form, of the hyperbola with center (3,2), vertex at (3,4) and focus at (3,-3)
8) Find the asymptotes of the following hyperbolas
2
2
a) x  y  1
16 4
2
2
b) ( y  1)  ( x  3)  1
4
25
9) Find the center, foci and vertices of the following ellipses
a) 3x2+4y2-36x + 32y+160 = 0
2
2
b) ( x  3)  ( y  1)  1
36
9
10) Find the equation, in the standard form, of the ellipse with foci at (4,0) and (4,-6) whose the major axis has the length of 10.
Graph the ellipse.
11) Write the equation for the graph below
12) Solve using the elimination method
2 x 2  y 2  4
 2
3x  2 y 2  6
13) Graph the equation
y 2 x2

1
9
4
2
2
14) Graph the equation ( x  1)  ( y  1)  1
9
25
15) Graph the equation 36x2 = 4y2+ 144
16) Solve using the elimination method
6 x  3 y  36

2 x  6 y  40
17) Find values of the following determinants
a) 3  2
1 4
1
2 3
b)  2 3
1
3 1 2
18) Graph the equation 4x2 + 9y2 = 36 and find its foci.
19) Solve for x
8 2
x 3
7
20) Graph the equations in the given system to determine the number of solutions. Then solve the system to find points of intersection.
2
2
a)  x  y  9
 y  x 2  3
2
2
b)  x  y  4
2
2

y  x  1
21) Write the equation, in the standard form, of the hyperbola whose graph is given below.
22) Check whether Cramer’s Rule can be used to solve the systems below. If yes, use Cramer’s Rule to solve them.
x  y  4z  2
a) 2 x  z  5

 x  y  4 z  3

b) 6 x  3 y  6

 4 x  y  10
c) 2 x  4 y  5

 4 x  8 y  10
x  4z  1
d) 2 x  y  6 z  4

2 x  3 y  2 z  8

23) Solve the system of equations. If the system has no solutions, say it is inconsistent. If the system has infinitely many solutions,
express them using set notation.
a)  x  5 y  2
 3x  15 y  6
b) 2 x  y  9
6 x  3 y  5
x  y  z  0
c)  x  2 y  3z  3

2 x  3 y  4 z  3

x  2 y  z  1
d) 2 x  3 y  4 z  5

3x  6 y  3z  2

24) Check whether (-4,1) is a solution of the system 2 x  15 y  7
10 x  42 y  22
25) Find the equation of the hyperbola with center at (0,0), focus at (2 5 ,0) and vertex at (4,0). Graph it.
26) Find the equation of the ellipse with center at (0,0), focus at (-5,0) and vertex at (8,0)
27) Solve the system of equations
 x  y  5 z  15

4 x  z  4
 x  5 y  z  29

28) Solve the system of equations. [Hint: Let u = 1 and v = 1 , and solve for u and v.]
x
y
2
x 


1 
 x
3
 18
y
2
 5
y
Answers:
1)a) (-2,-4), (-4, -2); b) (5,1), (-8, -11/2)
2) (38/7, 16/7)
3)



4)a) center (1,-3); vertices: (-1,-3),(3,-3); foci: 1  5,3 , 1  5,3 ;transverse axis: y = -3; asymptotes: y   1 x  5 , y  1 x  7
2
2
2
b) center (0,0); vertices: (0,-4),(0,4); foci: 0,5, 0,5 ;transverse axis: y = 0; asymptotes: y   4 x



3
c) center (-2,3); vertices: (-4,3),(0,3); foci:  2  13,3 ,  2  13,3 ;transverse axis: y = 3; asymptotes: y   3 x, y  3 x  6
2
2
5) y  x  1
16 64
2
2
6) x  y  1
25 39
2
2
(
7) y  2)  ( x  3)  1
4
21
1
2
1
2
11
8) a) y   x ; b) y   x  , y  x 
2
5
5
5
5
9) a) center: (6,-4); foci: (5,-4),(6,-4); vertices (4,-4).(8,-4)
b) center: (-3,-1); foci:  3  3 3,1 ,  3  3 3,1 ; vertices (-9,-1),(3,-1)

2
2
10) ( y  3)  ( x  4)  1
25
16
2
2
11) ( x  2)  ( y  1)  1
16
4
12) ( 2 ,0), ( 2 ,0)
13)


2
2
2
14)
15)
16) (8,-4)
17) a) -10; b) 42
18) foci ( 5 ,0), ( 5 ,0)
19) 8.5
20) a) 3 solutions; (0,-3), ( 5 ,2), ( 5 ,2)
b) 4 solutions;



3 / 2, 5 / 2 ,  3 / 2, 5 / 2 ,

3 / 2 , 5 / 2 ,  3 / 2 , 5 / 2

2
2
21) x  y  1
9 16
22) a) can’t be used; b) (6,14); c) can’t be used; d) (5,0,1)
23) a) {(x,y)| x = -5y+2, y is a real number}; b) inconsistent; c) {(x,y,z)| x = 3-z, y = -3+2z, z is any real number}; d) inconsistent
24) no
2
2
25) x  y  1
16
4
2
2
26) x  y  1
64
39
27) (0,5,4)
28) (1/3, ¼)