This document contains a 40 question multiple choice midterm exam on pre-calculus topics including conic sections (circles, ellipses, parabolas, hyperbolas), their standard forms and properties. Questions ask students to identify equations, graphs, centers, foci, vertices and solutions based on given geometric figures and algebraic expressions.

1. MIDTERM EXAMINATION in
Pre Calculus
Name: _____________________________________________ Date: ______________ Score: _______
Directions: Read each question carefully. Encircle the letter of your answer.
1. What type of conic section is being described by the equation?
𝑦 =
1
2
𝑥2
A. Circle B. Ellipse
C. Hyperbola D. Parabola
2. Which among the graph describes the given equation?
2 2
x y 25 
A. B.
C. D.
For items 3 and 4, use the figure below:
3. Where is the center of the circle located?
A. (1, -2) B. (2, -2) C. (2,2) D. (-2,2)
4. What is the equation of the circle in standard form?
A. 2 2
4 4 4 0x y x y     C. 2 2
4 4 4 0x y x y    
B. 2 2
4 4 4 0x y x y     D. 2 2
4 4 4 0x y x y    
5. Based on the given information, what is the standard form of the circle?
center: (15,5) and radius: 1
A. (x−15)2
+ (y + 5)2
= 1 B. (x + 15)2
+ (y + 5)2
= 1
C. (x−15)2
+ (y + 5)2
= 9 D. (x−15)2
+ (y−5)2
= 1
6. What is the center and radius of the given circle?

2. 2 2
2y y 18x 57 x    
A. center: (−9,1); radius: 5 B. center: (−10,0); radius: 5
C. center: (9,1); radius: 5 D. center: (2,10); radius: 5
7. If both x2
and y2
appear, and their coefficients A and B have different signs, then these
describes a:
A. Parabola B. Ellipse C. Circle D. Hyperbola
8. If both x2
and y2
appear, and their coefficients A and B have the same signs and are equal,
then these describes a:
A. Parabola B. Ellipse C. Circle D. Hyperbola
9. Which among the graph describes the given equation?
𝑥2
+
𝑦2
9
= 1
A. B.
C. D.
For items 10-13, use the figure below.
10. Where is the center of the ellipse located?
A. (-2,0) B. (-1,0) C. (2,0) D. (-1,0)
11. What is the length of the covertex of the ellipse from its center?
A. 2 units B. 3 units C. 4 units D. 5 units
12. How far is the second foci from the first foci?

3. A. 2 units B. 4 units C. 8 units D. 12 units
13. What the general equation of the given ellipse?
A.
 
2 2
1
1
16 9
x y
  C.
 
22
1
1
16 9
yx 
 
B.
 
2 2
1
1
9 16
x y
  D.
 
22
1
1
9 16
yx 
 
14. Where are the foci of the ellipse
2 2
x y
1
144 169
  ?
A. (5,0), (-5,0) B. (0,5), (0, -5) C. (1,7), (1, -3) D. (-1,6), (-1, -4)
15. Given the equations 2
x y and 2
y x , in how many points will the curves intersect?
A. 0 B. 1 C. 2 D. 3
16. From (12), where will the given curves intersect?
A. (0,0) B. (0,0),(1,1) C. (0,0),(1,1),(-1,-1) D. None of these
17. Given the equations 2 2
x y 16  and, in how many points will they intersect?
A. 0 B. 1 C. 2 D. 3
18. From (10), where will they intersect?
A. (4,0) B. (4,0),(0,-4) C. (-4,0),(4,0),(0,-4) D. None of these
19. In how many points can a circle and a line intersect?
A. 0 B. 1 C. 2 D. All of the above
20. Describe the graph of the equation  
2
y 1 4y   .
A. Parabola opening upward C. Parabola opening to the right
B. Parabola opening downward D. Parabola opening to the left
21. Determine the center of the circle given the equation   
2 2
x 2 y 1 1    .
A. (2,-1) B. (-2,-1) C. (-2,-1) D. (2,1)
22. Which of the following is an equation of a hyperbola?
A.
2
2
25
4
x
y  C. 2
3 5 6 0x y x   
B. 2 2
4 3 4 6 0x y x y     D.
2 2
1
4 9
x y
 
23. Which among the graph describes the given equation?
2 2
y y
1
16 16
 
A. C.

4. B. D.
24. What are the vertices of the given hyperbola?
 
2
2
x 4 y
1
9 36

 
A. (-1,0),(-7,0) B. (0,7),(0,1) C. (-4,3),(-4,-3) D. (-4,6),(-4,-6)
25. In the conic
2 2
x y
1
25 100
  , how far are the vertices from each other?
A. 10 B. 15 C. 20 D. 25
26. Which among the graph describes the given equation?
2
y x
A. B.
C. D.
27. Where is the focus of the given parabola 21
y x 6x 20
3
   ?
A.
33
, 7
4
 
  
 
B.
35
8,
4
 
 
 
C.
55
9,
8
 
  
 
D.
25
9,
4
 
  
 
28. Where is the vertex of the given parabola 2
y 3x 48x 198    ?
A. (-8,6) B. (-8,-6) C. (6,8) D. (6,-8)
29. What is the solution of the given system of equations?
2
2y 27x y 17 0
3x y 1 0
    
   

5. A. (1,2) B. (9,1) C. No solution D. (1,2), (-10,4)
30. Which of the following is not a conic section?
A. Parabola B. Circle C. Line D. Ellipse
31. What is the solution of the given system of equations?
2 2
x y 3x 7y 18 0
x y 3 0
    
  
A. (-3,0),(-6,3) B. (-2,-1) C. (-2,-1),(-6,3) D. (-3,0),(-2,-1)
For items 29-31, given the system of nonlinear equations below:
2 2
25
2 3 6
x y
x y
  

  
32. What kind of conic section is 2 2
1x y  ?
A. Circle B. Parabola C. Ellipse D. Hyperbola
33. What is the graph of the equation 2 3 6x y   ?
A. Circle B. Line C. Parabola D. Ellipse
34. Which of the following is a solution to the system?
A. (-3,0) B. (3,4) C. (-4,2) D. (2,3)
For items 35-36, use the figure below.
35. What are the point of intersection of the line and the circle?
A. (1,0) & (0, -1) C. (-1,0) & (0,1)
B. (-1,0) & (0,-1) D. (1,0) & (-1,0)
36. Which of the following are the equations of the line and the circle.
A.
2 2
1
1
x y
y x
  

 
C.
2 2
1
1
x y
x y
  

 
B.
2 2
1
1
x y
x y
  

 
D.
2 2
1
1
x y
y x
  

  

6. For items 37-39, refer to the figure below.
37. Where is the center of the hyperbola located?
A. (0,0) B. (0, -1) C. (0,1) D. (-1,0)
38. What are the asymptotes of the hyperbola?
A. y x  B. 2y x  C. 3y x  D. 4y x 
39. Where are the vertices of the the hyperbola?
A. ( 2, 1),(2, 1)   C. ( 1, 1),(1, 1)  
B. (2, 1),(1, 1)  D. ( 2,1),(2, 1) 
40. What is the general equation of the hyperbola?
A.  
22
1 1x y   C.  
2 2
1 1x y  
B.  
2 2
1 1x y   D.  
22
1 1x y  