Analytic Geometry

Question 1

Set 1 Problem 10: Find the general equation of the line with slope 3/4 and passes through (-2,7).

Answer 1

3x-4y+34=0 (p.9)

Question 2

Set 1 Problem 12: The ellipse has its center at (0,0), one end of minor axis at (5,0) and distance between foci 6.633. Find the equation of the ellipse.

Answer 2

36x²+25y²=900 (p. 9)

Question 3

Set 1 Problem 15: Find the equation of the line through (2,8) and perpendicular to x-2y+8=0.

Answer 3

2x+y=12 (p. 9)

Question 4

Set 1 Problem 17: Find the area of a polygon with vertices at (-5,0), (2,-4), (5,2), (3,5), and (0,3).

Answer 4

43.5 (p.10)

Question 5

Set 1 Problem 18: Find the general equation of the circle with center at (-3,6) and touches the x-axis.

Answer 5

x²+y²+6x-12y+9=0 (p.10)

Question 6

Set 1 Problem 25: Find the equation of the line with x-intercept 5 and y-intercept of -3.

Answer 6

3x-5y=15 (p.11)

Question 7

Set 1 Problem 26: The line with y=mx+3 passes through (2,1). Find the slope m.

Answer 7

-1 (p.11)

Question 8

Set 1 Problem 27: Find the distance between the parallel lines 3x-4y+4=0 and 3x-4y-16=0.

Answer 8

4 (p.11)

Question 9

Set 1 Problem 34: Given the equation of the circle x²+y²-8x-12y+2=0. Find the center of the circle.

Answer 9

(4,6) (p.12)

Question 10

Set 1 Problem 39: Given the equation of the parabola x²+16x+8y+40=0. Find the coordinates of its vertex.

Answer 10

(-8,3) (p. 13)

Question 11

Set 1 Problem 46: Find the area bounded by the parabola x²=8y and its latus rectum.

Answer 11

10.67 (p.13)

Question 12

Set 2 Problem 35: Find the general equation of the line through (3,-1) and parallel to the line x-3y+6=0.

Answer 12

x-3y=6 (p.28)

Question 13

Set 2 Problem 50: An ellipse has an equation x²+4y²=8. Find the equation of the diameter of the ellipse which bisects a system of parallel chords with slope 2/3.

Answer 13

3x+8y=0 (p.31)

Question 14

Set 3 Problem 11: What are the rectangular coordinates corresponding to the cylindrical coordinates (2, 2π/3, 3)?

Answer 14

(-1, √3, 3) (p.42)

Question 15

Given the following equations in polar form. Convert each to rectangular form.
Set 3 Problem 12: r=5sec⁡θ

Answer 15

x=5 (p.43)

Question 16

Given the following equations in polar form. Convert each to rectangular form.
Set 3 Problem 14: r=4

Answer 16

x²+y²=16 (p.43)

Question 17

Given the following equations in polar form. Convert each to rectangular form.
Set 3 Problem 15: r = 2/(1+sin⁡θ)

Answer 17

x²+4y-4=0 (p.43)

Question 18

Set 3 Problem 18: Find the volume of the solid bounded by the paraboloid y=x²+z² and the plane y=4.

Answer 18

8π (p.44)

Question 19

Set 3 Problem 27: Find the distance between the parallel lines 2x–5y+10=0 and 4x-10y-7=0.

Answer 19

1.207 (p.46)

Question 20

Set 3 Problem 38: Find the equation of the locus of a point which moves so that the sum of its distances from (2,0) and (-2,0) is always 8.

Answer 20

3x^2+4y^2=48 (p.48)

Question 21

Set 3 Problem 43: Compute the length of the latus rectum of the conic whose polar equation is rcos^2=2sinθ

Answer 21

2 (p.49)

Question 22

Set 3 Problem 58: Given the equation of the parabola with focus at (1,-1) and directrix y=-2.

Answer 22

x²+2x-6y-2=0 (p.52)

Question 23

Set 3 Problem 62: Given the equation of the circle x^2+y^2- 8x-12y+2=0. Find the area of the circle.

Answer 23

157.08 (p.53)

Question 24

Set 4 Problem 1: The ceiling in a hallway 10m wide is in the shape of a semi-ellipse and is 9m high at the center and 6m high at the sidewalls. Find the height of the ceiling 2m from either wall.

Answer 24

8.4m (p.60)

Question 25

Set 4 Problem 27: A triangle has vertices at A(1,3), B(7,0), and C(4,6). Locate the coordinates the orthocenter.

Answer 25

(3,4) (p.63)

Question 26

Set 4 Problem 36: Find the equation of the bisector of the acute angle formed by the intersection of the lines x+y=4 and 7x-y=4.

Answer 26

3x+y=6 (p.65)

Question 27

Set 4 Problem 62: What conic is represented by the equation 9x^2-24xy+6y^2-40x-30y=0?

Answer 27

hyperbola (p.69)

Question 28

Set 4 Problem 63: The slope of a line joining a moving point (1,-5) is always twice the slope of the line joining it to the origin. Find the equation of the locus of the moving point.

Answer 28

xy+5x+2y=0 (p.69)

Question 29

Set 4 Problem 64: Determine the polar equation of the parabola with vertex at (-2,0) and x+4=0 as the directrix.

Answer 29

4/(1-cosθ) (p.69)

Question 30

Set 5 Problem 28: A sphere has a radius of 3, and tangent to all three coordinate planes. Find the equation of the sphere if the center is in the first octant.

Answer 30

x²+y²+z²-6x-6y-6z+18=0 (p.82)

Question 31

Set 5 Problem 54: The area enclosed by the ellipse 4x^2+9y^2=36 is revolved about the line x=3. Find the volume of the solid generated.

Answer 31

36(π)^2 cubic units (p.85)

Question 32

Set 5 Problem 63: Find the equation of the line containing the point (4,-1) and satisfying the condition that the segment intercepted between the axes on the 4th quadrant is equal to 2sqrt(17).

Answer 32

x-4y=8 (p.86)

Question 33

Set 5 Problem 71: Find the distance from the point (1,4,6) to the plane x-3y+5z+8=0.

Answer 33

4.56 (p.89)

Question 34

Set 6 Problem 33: Find the angle between the planes 3x-y+z-5=0 and x+2y+2z+3=0.

Answer 34

72.45° (p.98)

Question 35

Set 6 Problem 60: A point moves so that its distance from the point (3,5) is always equal to its distance from the line y=1. Find the eccentricity of the curve.

Answer 35

1 (p.104)

Question 36

Set 7 Problem 23: A circle has its center (3,-3) and one end of a diameter is at P_1(2,-4). Find the other end P_2(x_2,y_2) of this diameter.

Answer 36

(4,-10) (p.114)

Question 37

Set 7 Problem 26: In 1990, the average annual cost of tuition and fees at 4-year colleges in the U.S. was approximately $1,980. In 2000, the average annual cost of tuition and fees was $3510. Let y be the average annual cost and x is the number x is the number of years after 1990. Write a linear equation that models the cost of tuition and fees for any given year after 1990.

Answer 37

y=153x+1980 (p.115)

Question 38

Set 7 Problem 35: A parabola has an equation x^2=4y. Find the equation if the diameter which bisects a system of chords parallel to the line x-2y=10.

Answer 38

1 (p. 116)

Question 39

Given the equation of the ellipse: 25x²+9y²-250x-54y+481=0. Set 7 Problem 51: Locate the center of the ellipse.

Answer 39

(5,3) (p.119)

Question 40

Given the equation of the ellipse: 25x²+9y²-250x-54y+481=0. Set 7 Problem 52: Compute the second eccentricity.

Answer 40

1.33 (p.119)

Question 41

Given the equation of the ellipse: 25x²+9y²-250x-54y+481=0. Set 7 Problem 53: Compute the length of the latus rectum.

Answer 41

3.6 (p.119)

Question 42

Set 7 Problem 55: The area of the ellipse x²+4y²=16 in the first quadrant is revolved a full revolution about the line x+5=0. Compute the volume of the solid generated.

Answer 42

264.43 cubic units (p.119)

Question 43

Set 7 Problem 69: Find the volume of a solid bounded by the coordinate planes and the plane 3x+4y+6z=24.

Answer 43

32 (p.122)

Question 44

Given the equations of two A and B: circle A: x^2+y^2=64 and circle B: x^2+y^2-16x=0. Set 8 Problem 30: Find the center of circle B.

Answer 44

(8,0) (p.133)

Question 45

Given the equations of two A and B: circle A: x^2+y^2=64 and circle B: x^2+y^2-16x=0. Set 8 Problem 31: Find the radical axis of the two circles.

Answer 45

x=4 (p.133)

Question 46

Given the equations of two A and B: circle A: x^2+y^2=64 and circle B: x^2+y^2-16x=0. Set 8 Problem 32: Find the length of the common chord of the two circles.

Answer 46

13.86 (p.133)

Question 47

Set 8 Problem 42: Compute the eccentricity of the hyperbola x^2-y^2=4.

Answer 47

1.414 (p.135)

Question 48

Given the equation of the ellipse x^2+4y^2=16. Set 8 Problem 43: Find the area of the ellipse.

Answer 48

8pi (p.135)

Question 49

Set 8 Problem 59: An ellipse has its major axis on the x-axis, center at (-3,0), one of the vertices at (-5,0) and length of latus rectum equal to 1. Find its equation.

Answer 49

(x + 3)^2+4y^2=4 (p.138)

Question 50

Set 9 Problem 15: Divide the line segment connecting the points A(2,0) & B(8,4) into 4 equal parts. Find the dividing point nearest to B.

Answer 50

(13/2,3) (p.147)

Question 51

Set 9 Problem 62: Find the equation of the line through (2,-5) and parallel to x-3y=8.

Answer 51

x-3y=17 (p.154)

Question 52

Set 9 Problem 63: Find the distance from the point (2,-5) to the line x-3y=8.

Answer 52

sqrt(10) (p.154)

Question 53

Set 9 Problem 64: Find the equation of the plane through the points (4,-2,-3), (5,4,1), and (0,6,5).

Answer 53

2x-3y+4z-2=0 (p.155)

Question 54

Set 9 Problem 66: Find the equation of the plane through the point (1,3,7) and parallel to the plane x-3y+2z=8.

Answer 54

x-3y+2z-2=6 (p.155)

Question 55

Set 9 Problem 68: An equilateral hyperbola xy=12 has the coordinate axes as asymptotes. Compute the length of the latus rectum.

Answer 55

9.8 (p.156)

Question 56

Set 10 Problem 10: Find the equation of a hyperbola with center at the origin, focus at (0,3), and y-2=0 as the directrix.

Answer 56

y^2-2x^2=6 (p.163)

Question 57

Set 10 Problem 15: The equation 2x^2-4xy+7x=10 represents a/an

Answer 57

hyperbola (p.164)

Question 58

Set 10 Problem 30: Find the distance between P1(3,120°) and P2(4,30°).

Answer 58

5 (p.167)

Question 59

Set 10 Problem 31: A paraboloid of revolution is generated by revolving about the x-axis the parabola z^2=4x which lie on the x-z plane. Find its equation.

Answer 59

y^2+z^2=4x (p.167)