Math 161 Review for Exam 6d
1. Write an equation for
(a) a parabola opening up with vertex ( 3, 2) and the point (1, 5).
(b) a parabola opening right with vertex (2, 8) and the point (3, 9).
(c) a circle with center (2, 3) and radius 4.
(d) a circle with center ( 1, 2) and radius 5.
(e) an ellipse with center (4, 1), vertices (1, 1) and (7, 1) and points on minor axis (4, 0) and (4, 2).
(f) an ellipse with center (2, 7), vertices (2, 1) and (2, 13) and points on minor axis (5, 7) and (1, 7).
(g) a hyperbola with center (2,  4), vertices (1, 4) and (5,  4) and whose asymptotes have slope  2 3 .
(h) a hyperbola with center (3, 5), vertices (3, 7) and (3, 3) and whose asymptotes have slope  1.
(i) a pair of lines through the point (4, 3) with slope  2 3 .
(j) a pair of lines through the point (2, 7) with slope  1.
(k) the point (2,  1).
(l) the point (5, 3).
(m) no solution.
2. Sketch the graph of each of the above equations and show the above features.
3. Write the equation in a standard form and identify the shape. Where applicable also identify the
center, radius, vertices, and asymptotes.
(a) x 2  y 2  8 x  6 y  16  0
(c) x 2  2 x  4 y 2  32 y  63  0
(e) x 2  4 x  4 y  16  0
(g) x 2  y 2  10 x  6 y  34  0
(i) y 2  2 y  8 x  17  0
(b) 9 x 2  18 x  4 y 2  24 y  9  0
(d)  x 2  4 y 2  2 x  24 y  31  0
(f) 16 x 2  64 x  9 y 2  90 y  145  0
(h) x 2  y 2  6 x  8 y  26  0
(j)  4 x 2  5 y 2  30 y  16 x  9  0
(k) x 2  y 2  2 x  8 y  8  0
(l) 4 x 2  y 2  4 x  2 y  2  0
4. Sketch the graph of each of the above equations and show the above features.
Math 161 Review 6d Answers
1.(a) x  3  16 3  y  2
2
(d) x  1   y  2  25
(b)  y  8  x  2
2
x  42   y  12
1
9
x  22   y  42  1
 y  52  x  32  1
(g)
(h)
9
4
4
4
2
2
2
2
(j) x  2    y  7   0
(k)  x  2    y  1  0
(m) Answers may vary. Some possible answers are
2

x  4
2
2
2
x  1   y  2  25 ,
  y  1  1 ,
9
2
2
(e)
(c) x  2   y  3  16
2
 x  2 2   y  7 2
1
9
36
x  42   y  32  0
(i)
9
4
2
2
(l) x  5   y  3  0
(f)
x  22   y  7 2
9
36
2.
(a)
(d)
(b)
(c)
(e)
2
 16
Math 161 Review 6d Answers
2. (f)
page two
(g)
(h)
(i)
(j)
(k)
(l)
(m)
3.(a) x  4    y  3  9 , circle with center (4,  3) and radius 3.
2
x  12   y  32
2
 1 , ellipse with center (1, 3) and vertices (1, 0) and (1, 6).
4
9
(c) y  4  12 x  1 a pair of lines with slope  1 2 through the point (1, 4).
(b)
Math 161 Review 6d Answers
3.(d)
2

x  1
 y  3 
2
4
page three
 1 , hyperbola with center (1, 3), vertices (1, 2) and (1, 4),
and asymptotes y  3  12 x  1 .
(e)  x  2   4 y  3 , parabola opening up with vertex (2, 3).
2
 x  2 2   y  5 2
 1 , ellipse with center (2, 5), vertices (2, 1) and (2, 9).
9
16
2
2
2
2
(g) x  5   y  3  0 , the point (5,  3).
(h)  x  3   y  4  1 , no real solution.
(f)
(i)  y  1  8 x  2 , parabola opening to the right with vertex (2, 1).
2
(j)
 y  32  x  22
4
5
 1 , hyperbola with center (2, 3), vertices (2, 1) and (2, 5),
and asymptotes y  3  2 5  x  2  / 5 .
(k) x  1   y  4  25 , circle with center (1, 4) and radius 5.
2
(l)  x  1 2  
2
2
 y  12
4
 1 , ellipse with center
12 ,1 and vertices 12 ,3 and 12 ,1 .
4.(a)
(c)
(b)
(d)
Math 161 Review 6 Answers
page four
4.(e)
(f)
(g)
(i)
(j)
(k)
(l)