Expert: Richard J. Raridon - 3/31/2010

Question
QUESTION: Find the distance between the pair of points:

(9, 1) and (2, -1)

d = 9.9  d = 7.28
d = 7  d = 21.17

2. Find the distance between the pair of points:

(0.5,-1) and (-1,1)

d = 3.54  d = 3.55
d = 2.5  d = 0.5

3. Find the distance between the pair of points:

(3.45, 0.32) and (1.45, 3.15)

d = 3.47  d = 2.45
d = 6.93  d = 2.83

4. Find the distance between the pair of points:

(-a,b) and (2a,4b)

ab  3Ö a2 + b2

3ab  a2 + b2

5. Find the midpoint of the line segment joining the points:

(13,6) and (0,6)

(6.5,0)  (13,12)
(6,6.5)  (6.5,6)

6. Find the midpoint of the segment joining the points:

(0.5,-1) and (-1,1)

(-2, 0)  (-0.25, 0)
(-0.5, 0)  (0.25, 0)

7. Find the midpoint of the segment joining the two points:

(3.45, 0.32) and (1.45, 3.15)

(2.45, 1.735)  (4.90, 3.46)
(1.735, 0.87)  (3.46, 4.90)

8. Find the midpoint of the line segment joining the two points:

(-a, b) and (2a, 4b)

(1.5a, 2.5b)  (0.5a, 1.5b)
(0.5a, 2.5b)  (2a, 0.4b)

9. Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(-4, 3) and M(0, 0)

(0, 3)  (1, -4)
(0, 0)  (4, -3)

10. Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(6, -2) and M(0, 5)

(4, 8)  (6, -12)
(-6, 12)  (12, -6)

11. Find the coordinates of Q given that M is the midpoint of line segment PQ:

P(0, 0) and M(h, k)

(2h, 2k)  (h, k)
(-2k, -2h)  (2h, k)

12. Find the midpoint of the segment joining the two points:

(Ö 2 , 1)
and(-Ö 2 , 0)
(0, 0.5)  (0.5, 0)
(Ö 1 , -2)
 (Ö 2 , -1)


13. Find the midpoint of the segment joining the two points:

(0.5, -1) and (-1, 1)

(-0.25, 0)  (0, -0.25)
(0.25, 0)  (0, 0.25)

14. For the given vertices what type of triangle is formed:

Vertices:  A(4, -5), B(-2, -8), C(-8, 4)

Right  Obtuse
Isosceles  Equilateral

15. For the given vertices what type of triangle is formed:

Vertices:  A(2, -3), B(2, 3), C(5, 0)

Isoceles right triangle  Isosceles
Equilateral  Scalene right triangle

16. What is the area of the triangle formed with the given vertices:

Vertices: A(2,-3), B(2,3), C(5,0)

9  18
81  27

17. Find an equation of the perpendicular bisector of line segment AB:

A(2, 1), B(-2, 3)

y = 2x - 2  y = 2x + 2
y = 4x - 2  y = 4x + 2

18. Find an equation of the perpendicular bisector of line segment AB:

A(-9, -3), B(1, -7)

y = 2x + 5  y = 2.5x + 15
y = 2.5x + 5  y = 0.4x + 15

19. Find the distance from any point (x,y) to the origin: Ö x + Ö y  
 x + y
Ö x2 + y2
 Ö x2  + Ö y2  


20. Find an equation of the circle with the center and radius:

(-3, 1); r = 5

(x - 3)2 + (y + 1)2 = 25  (x + 3)2 + (y - 1)2 = 25
(x - 3)2 + (y + 1)2 = 5  (x + 3)2 + (y + 1)2 = 5

21. Find an equation of the circle with the center and radius:

(-5,3); r = 0.167

(x + 5)2 + (y - 3)2 = 0.028  (x + 5)2 + (y - 3)2 = 0.167
(x - 5)2 + (y + 3)2 = 0.167  (x - 5)2 + (y + 3)2 = 0.028

22. Find the center and radius of a circle with equation:

x2 + y2 - 81 = 0

C(1, 1); r = 3  C(0, 0); r = 3
C(1, 1); r = 9  C(0, 0); r = 9

23. Find the center and radius of a circle with equation:

x2 + y2 + 10x - 4y + 2 = 0

C(-5, 2); r = 3Ö 3
 C(-5, 2); r = 2Ö 3

C(5, -2); r = 9  C(5, -2); r = 3

24. Find the center and radius of a circle with equation:

x2 + y2 - 5y + 4 = 0

C(0, 2.5); r = 1.5  C(0, -2.5); r = 2.25
C(0, 2.5); r = 2.25  C(0, -2.5); r = 1.5

25. Find the center and radius of a circle with equation:

4x2 - 16x + 4y2 - 24y + 36 = 0


C(2, 3); r = 2  C(2, 3); r = 3
C(3, 2); r = 2  C(3, 2); r = 3

26. Find the equation of a circle:

center (-2, 0); passes through (2, 0)

(x + 2)2 + y2 = 16  (x - 2)2 + y2 = 4
(x + 2)2 + (y + 0)2 = 4  (x - 2)2 + y2 = 16

27. Find the equation of the circle:

center on line x + y = 4; tangent to both first quadrant coordinate axes

(x + 2)2 + (y + 2)2 = 2  (x - 2)2 + (y - 2)2 = 2
(x + 2)2 + (y + 2)2 = 4  (x - 2)2 + (y - 2)2 = 4

28. If the directrix of a parabola is:

D: y = 2 and the vertex is V(2, 4), what is the focus?

F(-2, 6)  F(4, 0)
F(0, 4)  F(2, 6)

29. If the focus of a parabola is F(-3, -1) and vertex is V(1, -1), what is the directrix? D: x = 5  D: x = -1
D: x = -3  D: x = -5

30. If the directrix of a parabola is D: x = -7 and the vertex is V(1, 3), what is the focus? (-10, 5)  (9, 3)
(3, 5)  (-3, 10)

ANSWER: 1) 7.28  2) 2.5  3) 3.47  4) 3(a^2+b^2)^1/2  5) 6.5,6  6) -0.25,0  7) 2.45,1.735  8) 0.5a, 2.5b
9) 4,-3  10) -6,12  11) 2h,2k  12) 0,0.5  13) -0.25,0  14) right  15) isoceles right 16) 9
17) y=2x+2  18) y=2.5x+5  19) (x+y)^1/2  20) (x+3)^2+(y-1)^2 = 25  21) (x+5)^2+(y-3)^2 = 0.028
22) C(0,0); r=9  23) C(-5,2); r=3(3^1/2)  24) C(0,2.5); r= 1.5  25) C(2,3); r=2
26) (x+2)^2 +y^2 = 16  27) (x-2)^2 +(y-2)^2 = 4  28) 2,6  29) x=5  30) 9,3
I hope you learned something from all this

---------- FOLLOW-UP ----------

QUESTION: Find an equation of an ellipse having the given intercepts:

x-intercept: ±3

y-intercept: ±4

x2 + y2 = 1
3 4
 x2 - y2 = 1
9 16

x2 + y2 = 1
9 16
 x2 - y2 = 1
3 4


2. Find an equation of an ellipse having the given intercepts:


x-intercept: ±Ö 6


y-intercept: ±Ö 12
x2 + y2 = 1
6 12
 x2 - y2 = 1
6 12

x2 + y2 = 1
3 6
 x2 - y2 = 1
3 6


3. Find an equation of the ellipse having the given points as foci and the given sum of the focal radii

(-9, 0); (9, 0); 30

x2 + y2 = 1
15 12
 x2 + y2 = 1
225 144

x2 - y2 = 1
15 12
 x2 - y2 = 1
225 144


4. Choose the correct coordinates of the foci

x2 + 4y2 - 16 = 0

(12, 0) and (-12, 0)  (-2√3, 0) and (2√3, 0)
(-√12, 0) and (√12, 0)  (-4√3, 0) and (4√3, 0)

5. Find an equation of the hyperbola described:

Foci (0, -8) and (0, 8); difference of focal radii 10

y2 - x2 = 1
25 39
 y2 + x2 = 1
25 39

y2 - x2 = 1
5 39
 y2 + x2 = 1
5 39


6. Find an equation of the hyperbola described:

Asymptotes y = x and y = -x; foci (-4, 0) and (4, 0)

x2 - y2 = 1
8 8
 x2 + y2 = 1
4 4

x2 - y2 = 1
8 2
 x2 + y2 = 1
8 2


7. Identify the conic. Choose the conic and its center.

16x2 + 25y2 - 96x - 200y = -144

Ellipse (3, -4)  Ellipse (-3, 4)
Ellipse (3, 4)  Ellipse (-3, -4)

8. Identify the conic. Choose the conic and its center.

x2 - 4y2 + 6x - 8y = 11

Hyperbola (3, 1)  Hyperbola (-3, -1)
Hyperbola (4, 1)  Hyperbola (-4, 1)

9. Identify the conic. Choose the conic and its center.

16x2 - 9y2 + 64x + 18y + 199 = 0

Hyperbola (-2, 1)  Hyperbola (1, -2)
Hyperbola (-2, -1)  Hyperbola (-1, 2)

10. Write an equation, in standard form, of each circle described by the given conditions:

center: (0,0), radius: 12

x2 + y2 = 144  x2 + y2 = 24
2x2 + y2 = 144  2x2 + 2y2 = 24

11. Give the radius and the coordinates of the center of the circle:

(x - 1)2 + y2 = 144

center: (0, 0), radius: 12  center: (0, 0), radius: 72
center: (1, 0), radius: 12  center: (1, 0), radius: 72

12. Rewrite x2 - 4x + y2 + 8y - 16 = 0 in standard form. (x + 2)2 - (y - 4)2 = 4  (x - 2)2 - (y + 4)2 = 4
(x + 2)2 + (y - 4)2 = 36  (x - 2)2 + (y + 4)2 = 36

13. Find the area of an ellipse (πab) given an ellipse with equation 9x2 + 36y2 = 324.  18π  36π
9π  324π

14. Solve by completing the square:
x2 - 4x = -3

4, 2  1, 3
2, 3  0, 1

15. Determine whether the parabola is vertical or horizontal:
x2 = 4y

vertical  horizontal
both  cannot be determined

16. Determine whether the parabola opens up or down:
y/-6 = x2 +3

cannot be determined  both
up  down

17. Use the information to write an equation of the parabola:
Focus: (-8, 0), Directrix: x = 8

x2 = -8x  y2 = -8x
y2 = x/-2  y2 = -32x

18. Use the information to write an equation of the parabola:
Focus: (-8, -5), Vertex: (2, -5)

(y + 5)2 = -40(x - 2)  (x + 5)2 = 4(y - 2)
(y - 5)2 = -4(x - 2)  (x - 5)2 = 40(y - 2)

19. Identify the conic section whose equation is given:
y = x2

Hyperbola  Circle
Parabola  Ellipse

20. Identify the conic section whose equation is given:
4x2 - 16y2 = 16

Circle  Parabola
Hyperbola  Ellipse

21. Identify the conic section whose equation is given:
16x2 + 4y2 = 16

Circle  Parabola
Hyperbola  Ellipse

22. Identify the conic section whose equation is given:
(5y - x)(5y + x) = 100

Circle  Hyperbola
Ellipse  Parabola

23. Identify the conic section whose equation is given:
3x2 - 12x + y + 7 = 0

Parabola  Circle
Hyperbola  Ellipse

24. Identify the conic section whose equation is given:
5y2 + 50y + 275 = 100x - 2x2

Circle  Parabola
Hyperbola  Ellipse

25. Identify the conic section whose equation is given:
x2 + 10x + y2 + 12y = 60

Circle  Parabola
Ellipse  Hyperbola

Part 2
Select the best answer from the choices provided. (Each question is worth two points)
26. Choose the correct foci:

2x2 + y2 = 8

(±2, 0)  (0, ±4)
(0, ±2)  (0, 2)

27. Choose the correct foci:

5x2 + 9y2 = 45

(±2, 0)  (±4, 0)
(0, ±2)  (0, ±4)

28. Choose the correct foci:

9x2 + 4y2 =9

(0, ±2.5)  (0, ±1.1)
(±2.5, 0)  (±1.1, 0)

29. Choose the correct foci:

y2 = 5x2 + 25

(0, ±5.48)  (0, 7.75)
(±30, 0)  (0, ±30)

30. Choose the correct foci:

25x2 - 144y2 = 3600

(0, ±13)  (5, 12)
(12, 5)  (±13, 0)

Answer
1) x^2/9 +y^2/16 = 1  2) x^2/6 +y^2/12 = 1  3) x^2/225 +y^2/144 = 1  4) (-12^1/2,0),(12^1/2,0)
5) y^2/25 -x^2/39 = 1  6) x^2/8 -y^2/8 = 1  7) ellipse (3,4)  8) hyperbola (-3,1)
9) hyperbola (-2,1)  10) x^2+y^2 = 144  11) center (1,0)radius: 12  12) (x-2)^2+(y+4)^2 = 36
14) 1,3  15) vertical  16) down  17) y^2 = -32x  18) (y+5)^2 = -40(x-2)  19) parabola
20) hyperbola  21) ellipse  22) hyperbola  23) parabola  24) ellipse  25) ellipse
26) x^2/4 +y^2/8 = 1  27) (0,+/-2)  28) (+/-2.5,0)  29) (0,+/-5.48)  30) (0,+/-13)
I've worked enough for you.  Now you can do some.