Quadratic Relations Handout

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CIRCLE PROBLEMS  (Section 9.2, answers below)

For problems 1 through 12, complete the square (if necessary) to find the center and radius. Then sketch the graph.
1. x2  + y2 - 10x + 8y + 5 = 0                                          7. x2  + y2 + 4x - 5 = 0
2a.  x2  + y2 +12x - 2y + 21 = 0                                    8. x2  + y2 -14y + 48 = 0
2b. -2x2   -  2y2 - 24x + 4y - 42 = 0                        9. x2  + y2  = 49
3. x2  + y2 - 6x - 4y -  12 > 0                                   10. x2  + y2  = 4
4. x2  + y2 + 16x + 10y - 11 < 0                                    11. x2  + y2  = 0
5. x2  + y2 - 8x + 6y - 56 £ 0                                        12. x2  + y2  + 16 = 0
6. x2  + y2 + 4x - 18y + 69 ³ 0

 

For problems 13 through 18, write an equation of the circle described.
13. Center (7, 5), containing (3, -2) 16. Center (5, -4), containing (0, 3) 
14. Center (-4, 6), containing (-2, -3)                    17. Center at origin, containing (-6, -8) 

15. Center (-9, -2), containing the origin.            

18. Center at origin, containing (-5, 1) 

 

19. Write a general equation for the circle:
a. of radius r, centered at a point on the x-axis c. of radius r, centered at the origin.
b. of radius r, centered at a point on the y-axis.



ELLIPSE PROBLEMS  (Section 9.3, answers below)

For problems 1 through 16, after expressing in standard form:
a. Find center, major vertices, focal distance (from center), foci, and covertices (endpoints of minor vertex) , and eccentricity.
b. Sketch graph using data found in part a.
1. 4x2  + 9y2 - 16x + 90y + 205 = 0  9. 16x2  + 25y2 - 300y + 500 = 0
2. 4x2  + 36y2 +40x - 288y + 532 = 0 10. 36x2  + 9y2 - 216x = 0
3. 49x2   +  16y2 + 98x - 64y - 671 = 0 11.100x2  + 36y2  > 3600
4. 25x2  + 4y2 - 150x + 32y + 189 = 0 12. 25x2  + 49y2 £ 1225
5. x2  + 4y2 + 10x + 24y + 45 = 0     13. 12x2  + y2  = 48
6. 16x2  + y2 - 128x - 20y + 292 = 0 14. x2 + 6y2 = 25
7. 25x2  + 9y2 + 50x - 36y - 164 < 0 15. 5x2 + 8y2 = 77

8. 4x2  + 36y2 + 48x + 216y + 324  ³ 0

 16. 11x2 + 5y2 = 224

 

17. A construction company has a contract to build a football stadium in the form of two concentric ellipses, with the field inside the inner ellipse, and the seats between the two ellipses. The seats are in the intersection of the graphs of

       x2  + 4y2 ³ 100  and  25x2  + 36y2 £ 3600

where each unit of the graph represents 10 meters.

    a. Draw the graph of the seating area.
    b. The area of an elliptical region is pab, where a and b are the semi-axes. If each seat occupies 0.8 of a square
         meter, what is the seating capacity of the stadium?
18. Show that the equation of an ellipse, (x - h)2/a2 + (y - k)2/b2 = 1,  reduces to the equation of a circle if a =  b. 




PARABOLA PROBLEMS  (Section 9.5, answers below)

After expressing in standard form, find the vertex, p, focus, directrix, LR, endpoints of the latus rectum, and the AOS. Include these in sketching the graph of the parabola. 
1. x = y2 - 4y + 3 6. x = y2/3 - 2y - 9
2. x = y2 + 2y - 3   7. y = -4x2 + 20x - 16
3. x = -3y2 - 12y - 5 8. y = x2/5 + 2x - 11/5
4. x = -5y2 + 30y + 11 9. x = y2/4
5. x = y2/2 + 3y + 4 10.x = y2/10


CONIC SECTIONS  (Section 9.6, answers below)

Identify the graph defined by each relation. If the relation doesn't define a conic section, so state.
1. -3x2  - 3y2 - 12x + 15 = 0 5. 4x = -12y2 - 48y - 20
2. -3x2  - 3y2 - 12x - 15 = 0 6. x2  + y2 = 0
3. 8x2  + y2/2 - 64x - 10y + 146 = 0 7. -12y2 - 4x - 48y - 20 = 0
4. 16x2  - y2 = -16



9. 2 ANSWERS: 1. (x - 5)2 + (y + 4)2 = 36; C(5, -4), r = 6;  3. (x - 3)2 + (y - 2)2 > 25; C(3, 2),
r = 5;   5. (x - 4)2 + (y + 3)2 £ 81; C(4, -3), r = 9;  7. (x + 2)2 + y2 = 9; C(-2, 0), r = 3;
9. x2 + y2 = 49; C(0, 0), r = 7;  11. x2 + y2 = 0; C(0, 0), r = 0;  13. (x - 7)2 + (y - 5)2 = 65;
15. (x + 9)2 + (y + 2)2 = 85;  17. x2 + y2 = 100;  19a. (x - h)2 + y2 = r2;  19b. x2 + (y - k)2 = r2;
19c. (x - h)2 + (y - k)2 = r2


9. 3 ANSWERS: 1. (x - 2)2/32 + (y + 5)2/22 = 1, C(2, -5), c = Ö5;   V1(-1, -5), V2(5, -5),
F1(2-Ö5, -5), F2(2+Ö5, -5), CV1(2, -3), CV2(2, -7), e = (Ö5)/3
2. (x + 5)2/62 + (y - 4)2/22 = 1, C(-5, 4), c = 4Ö2;   V1(-11, 4), V2(1, 4),
F1(-5-4Ö2, 4), F2(-5+4Ö2, 4), CV1(-5, 6), CV2(-5, 2), e = (2Ö2)/3
3. (x + 1)2/42 + (y - 2)2/72 = 1, C(-1, 2), c = Ö(33), V1(-1, 9), V2(-1, -5),
F1(-1, 2+Ö(33)), F2(-1, 2-Ö(33)), CV1(-5, 2), CV2(3, 2), e = Ö(33)/7
4. (x - 3)2/22 + (y + 4)2/52 = 1, C(3, -4), c = Ö(21), V1(3, 1), V2(3, -9),
F1(3, -4+Ö(21)), F2(3, -4-Ö(21)), CV1(1, -4), CV2(5, -4), e = Ö(21)/5
 5. (x + 5)2/42 + (y + 3)2/22 = 1, C(-5, -3), c = 2Ö3, V1(-9, -3), V2(-1, -3),
F1(-5-2Ö3, -3), F2(-5+2Ö3, -3), CV1(-5, -1), CV2(-5, -5), e = (Ö3)/2
6. (x - 4)2/22 + (y - 10)2/82 = 1, C(4, 10), c = 2Ö(15), V1(4, 18), V2(4, 2),
F1(4, 10+2Ö(15)), F2(4, 10-2Ö(15)), CV1(2, 10), CV2(6, 10), e = (Ö15)/4
 7. (x + 1)2/32 + (y - 2)2/52 < 1,  C(-1, 2), c = 4, V1(-1, 7), V2(-1, -3),
F1(-1, 6), F2(-1, -2), CV1(-4, 2), CV2(2, 2), e = 4/5 
8. (x + 6)2/62 + (y + 3)2/22 >= 1, C(-6, -3), c = 4Ö(2), V1(-12, -3), V2(0, -3),
F1(-6-4Ö(2), -3), F2(-6+4Ö(2), -3), CV1(-6, -1), CV2(-6, -5), e = [2Ö2]/3
9. x2/52 + (y - 6)2/42 = 1, c = 3;    
11. x2/62 + y2/102 > 1, C(0, 0), c = 8, V1(0, 10), V2(0, -10),
F1(0, 8), F2(0, -8), CV1(-6, 0), CV2(6, 0), e = 4/5  
12. x2/72 + y2/52 > 1, C(0, 0), c = 2Ö6, V1(-7, 0), V2(7, 0),
F1(-2Ö6, 0), F2(2Ö6, 0), CV1(0, 5), CV2(0, -5), e = 2Ö(6)/7
13. x2/22 + y2/Ö(48)2 = 1, C(0, 0),  c = 2Ö(11), V1(0, 4Ö3), V2(0, -4Ö3),
F1(0, 2Ö(11)), F2(0, -2Ö(11)), CV1(-2, 0), CV2(2, 0), e = [Ö33]/6 
14. x2/52 + y2/(25/6)2 > 1, C(0, 0), c = [5Ö30]/6, V1(-5, 0), V2(5, 0),
F1(-[5Ö30]/6, 0), F2([5Ö30]/6, 0), CV1(0, [5Ö6]/6), CV2(0, -[5Ö6]/6), e = Ö(30)/6 
15.  x2/Ö(77/5)2 + y2/Ö(77/8)2 = 1,  c = Ö(231/40);


9. 5 ANSWERS: 1. (y - 2)2 = x + 1, V(-1, 2), p = 1/4, F(-3/4, 2), directrix: x = -5/4, LR = 1, 
L(-3/4, 5/2), R(-3/4, 3/2), AOS: y = -2, y-int = 1 and 3 x-int = 3;    
2. (y + 1)2 = x + 4, V(-4, -1), p = 1/4, F(-15/4, -1), directrix: x = -17/4, LR = 1, 
L(-15/4, -1/2), R(-15/4, -3/2), AOS: x = -1
3. (y + 2)2 =  (-1/3)(x  - 7), V(7, -2), p = -1/12, F(6 11/12, -2), directrix: x = 7 1/12, LR = 1/3,
L(6 11/12, -11/6), R(6 11/12, -13/6)   y-int approx -0.5, -3.5, x-int = -5, AOS: y = -2  
4. (y - 3)2 = (-1/5)(x  - 56), V(56, 3), p = -1/20, F(55 19/20, 3), directrix: x = 56 1/20, LR = 1/5,
L(55 19/20, 3.1), R(55 19/20, 2.9),  AOS: y = 3
5. (y + 3)2 = 2(x + 1/2), V(-1/2, -3), p = 1/2, F(0, -3), directrix: x = -1, LR = 2,
L(0, -5/2), R(0, -7/2), AOS: y = -3, y-int  = -2 and -4, x-int = 4;   
6.  (y - 3)2 = 3(x + 12), V(-12, 3), p = 3/4, F(-45/4, 3), directrix: x = -51/4, LR = 3,
L(-45/4, 9/2), R(-45/4, 3/2),  AOS: y = 3
7. (x - 5/2)2 = (-1/4)(y - 9), V(5/2, 9),  p = -1/16, F(5/2, 8 15/16), directrix: y = 9 1/16, LR = 1/4, 
L(19/8, 8 15/16), R(21/8, 8 15/16),  AOS: x = 5/2, y-int  = -16, x-int = 1 and 4;  
8. (x + 5)2 = 5(y + 36/5), V(-5, -36/5), p = 5/4, F(-5, -119/20), directrix: y = -169/20, LR = 5,
L(-15/2, -119/20), R(-5/2, -119/20), AOS: x = -5
9. y2 = 4x, V(0, 0), p = 1, F(1, 0), directrix: x = -1, LR = 4
L(1, 2), R(1, -2), AOS: y = 0, y-int = x-int = 0
10. y2 = 10x, V(0, 0), p = 5/2, F(5/2, 0), directrix: x = -5/2, LR = 10
L(5/2, 5), R(5/2, -5), AOS: y = 0, y-int = x-int = 0


9. 6 ANSWERS: 1. circle: C(-2, 0), r = 3;   2.  Doesn't define a conic section.
3.   Ellipse: vertical, C(4, 10), semi-minor = 2, semi-major = 8;   4. hyperbola: C(0, 0);
5. parabola: horizontal, V(7, -2); 6. point circle: C(0, 0); 7. parabola: horizontal, V(7,-2)


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