Analytic Geometry 2.pdf

ANALYTIC GEOMETRY 2 Conic Section

ENGR. REYNILAN L. DIMAL

CONIC SECTIONS Conics is the locus of a point which moves in such a way that the ratio of its distance from a fixed point (focus) and a fixed line (directrix) is constant. The constant ratio is called eccentricity.

CONIC SECTIONS Circle Cutting parallel to “base”

plane t he

CONIC SECTIONS Ellipse Cutting plane not parallel to any element of the circular cone

CONIC SECTIONS Parabola Cutting plane parallel to slant height of the circular cone

CONIC SECTIONS Hyperbola Cutting plane parallel to the axis

General Equation of the Second Degree Curves Ax2 + Bxy + Cy2 +Dx + Ey + F = 0 Relationship of A and C A=C, B=0 A  C, but the same sign If either A or C is zero A and C are opposite in signs

Type of Conic Section Circle Ellipse Parabola Hyperbola

General Equation of the Second Degree Curves Ax2 + Bxy + Cy2 +Dx + Ey + F = 0 B2 – 4AC

→Discriminant (D)

Behavior of Discriminant (D) D<0 D=0 D>0

Type of Conic Section Ellipse Parabola Hyperbola

REVIEW QUESTIONS 1. What conic section is represented by the equation 4x2 + 8x – y2 + 4y – 15 = 0 a. Circle

c. Parabola

b. Hyperbola

d. Ellipse

REVIEW QUESTIONS 2. What conic section is represented by the equation x2 + 4y2 + 4xy + 2x – 10 = 0? a. Circle

c. Parabola

b. Hyperbola

d. Ellipse

REVIEW QUESTIONS 3. Determine the nature of the curve: 4x2 + 3y2 – 8x + 16y + 19 = 0. a. Circle

c. Parabola

b. Hyperbola

d. Ellipse

CIRCLE The plane figure obtained as a locus of a point whose distance from a fixed point called the center is constant. Eccentricity is 0

CIRCLE General Equations: Ax 2  Ay 2  Dx  Ey  F  0 x  y  dx  ey  f  0 2

2

CIRCLE Standard Equations: Center at Any Point (h,k) ( x  h) 2  ( y  k ) 2  r 2

Center at Origin x y r 2

2

2

CIRCLE Standard Equations: x 2  y 2  dx  ey  f  0

Coordinate of the center e k  2

d h 2

Length of the Radius: r  h k  f 2

2

CIRCLE Standard Equations: Given Ends of Diameter

Equation 2 x  x )  2( y  2y )( y  y )  0 ( x  hx1))(  ( y 2 k )  r 1 2

REVIEW QUESTIONS 4. Find the equation of the circle whose center is at (3, -5) and whose radius is 4. a. x2 + y2 -6x + 10y + 18 = 0 b. x2 + y2 -6x - 10y + 18 = 0 c. x2 + y2 +6x + 10y + 18 = 0 d. x2 + y2 +6x - 10y + 18 = 0

REVIEW QUESTIONS 5. What is the center of the curve x2 + y2 -2x - 4y - 31 = 0. a. (-1, -2)

c. (1, -2)

b. (-1, 2)

d. (1, 2)

REVIEW QUESTIONS 6. What is the equation of a circle whose ends of diameter are (10, 2) and (6, -4). a. x2 + y2 -16x + 2y + 52 = 0 b. x2 + y2 -8x + 2y - 48 = 0 c. x2 + y2 +16x - 12y + 52 = 0 d. x2 + y2 +26x - 2y + 48 = 0

(10, 2)

(6, 4)

REVIEW QUESTIONS 6. What is the equation of a circle whose ends of diameter are (10, 2) and (6, -4). a. x2 + y2 -16x + 2y + 52 = 0 b. x2 + y2 -8x + 2y - 48 = 0 c. x2 + y2 +16x - 12y + 52 = 0 d. x2 + y2 +26x - 2y + 48 = 0

REVIEW QUESTIONS 7. Find the value of k for which the equation x2 + y2 + 4x - 2y - k = 0 represents a point circle. ECE Board Apr. 1998

a. 5 b. 6

c. -6 d. -5

PARABOLA Is the locus of a point which moves so that it is always equidistant from a fixed point called focus and a fixed line called directrix Eccentricity is equal to 1

PARABOLA General Equations:

Axis Parallel to the y-axis Ax 2  Dx  Ey  F  0

Axis Parallel to the x-axis Ay 2  Dx  Ey  F  0

PARABOLA Standard Equations:

Axis Vertical: Vertex (h,k) ( x  h) 2  4a( y  k )

Axis Horizontal: Vertex (h,k) ( y  k ) 2  4a( x  h)

PARABOLA Length of Latus Rectum

Latus Rectum - line that runs parallel to directrix and passes through the focus

REVIEW QUESTIONS 8. The parabola y=-x2 +x +1 opens:

a. to the left b. upward

c. to the right d. downward

REVIEW QUESTIONS 9. Compute the focal length and the length of latus rectum of parabola y2 + 8x – 6y +25 = 0 a. 2, 8 b. 16, 64

c. 4, 16 d. 1, 4

REVIEW QUESTIONS 10. A cable suspended from supports that are the same height and 600 feet apart has sag of 100 feet. If the cable hangs in the form of a parabola, find its equation. a. y2 = 900x b. y2 = 400x

c. x2 = 400y d. x2 = 900y

(300,100)

100

(0,0)

600

REVIEW QUESTIONS 10. A cable suspended from supports that are the same height and 600 feet apart has sag of 100 feet. If the cable hangs in the form of a parabola, find its equation. a. y2 = 900x b. y2 = 400x

c. x2 = 400y d. x2 = 900y

REVIEW QUESTIONS 11. An arch 18 m high has the form of parabola with a vertical axis. The length of a horizontal beam placed across the arch 8 m from the top is 64 m. Find the width of the arch at the bottom a. 86m b. 106m

c. 96m d. 76m

8 18

32

x

(32, 8) 32

x

(X, 18)

REVIEW QUESTIONS 11. An arch 18 m high has the form of parabola with a vertical axis. The length of a horizontal beam placed across the arch 8 m from the top is 64 m. Find the width of the arch at the bottom a. 86m b. 106m

c. 96m d. 76m

ELLIPSE Is the locus of a point which moves so that the sum of its distance from two fixed points (foci) is constant and is equal to the length of the major axis

ELLIPSE

ELLIPSE Standard Equations: General Equation 22 2 2 ( x  h ) ( y  k ) Ax  Cy  Dx  Ey  F  0  1 2 2 a b

Major Axis Horizontal ( x  h) ( y  k )  1 2 2 b a 2

2

Major Axis Vertical

ELLIPSE

Key Formulas

Length of Major Axis

2a

Length of Minor Axis

2b

ELLIPSE

Key Formulas

Length of Latus Rectum 2b 2 LR  a

Eccentricity c a

For an ellipse, eccentricity is lesser than 1

ELLIPSE

Key Formulas

Focal Distance (c) c  a 2  b2

c  ea

Distance From Center to Directrix (d) a d e

REVIEW QUESTIONS 12. Find the major axis of the ellipse x2 + 4y2 -2x – 8y + 1 = 0. a. 2 b. 4

c. 10 d. 6

REVIEW QUESTIONS 13. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to directrix is: ECE Board Nov. ‘95

a. 6.047 b. 6.532

c. 6.614 d. 6.222

REVIEW QUESTIONS 14. An ellipse has an eccentricity of 1/3. Compute the distance between the directrices if the distance between foci is 4.

a. 18 b. 36

c. 32 d. 38

REVIEW QUESTIONS 15. The major axis of the elliptical path in which the earth moves around the sun is approximately 186, 000, 000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. a. 93, 000, 000 miles b. 94, 335, 100 miles c. 91, 450, 000 miles d. 94, 550, 000 miles

REVIEW QUESTIONS 15. The major axis of the elliptical path in which the earth moves around the sun is approximately 186, 000, 000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. a. 93, 000, 000 miles b. 94, 335, 100 miles c. 91, 450, 000 miles d. 94, 550, 000 miles

HYPERBOLA Is the locus of a point which moves so that the difference of the distances from two fixed points (foci) is constant and is equal to the length of the transverse axis

HYPERBOLA General Equations Transverse Ax  Cy  Dx  Ey  F  0 axis horizontal 2

2

Transverse Cy  Ax  Dx  Ey  F  0 axis vertical 2

2

HYPERBOLA Standard Equations ( x  h) 2 ( y  k ) 2  1 2 2 a b

Transverse axis horizontal

( y  k ) 2 ( x  h) 2  1 2 2 a b

Transverse axis vertical

“a” is always the denominator of the positive term

HYPERBOLA Coordinate of the Center (h,k) D h  2A

E k  2A

HYPERBOLA

Key Formulas

Length of Latus Rectum 2b 2 LR  a

Eccentricity c a

For a hyperbola, eccentricity is greater than 1

HYPERBOLA

Key Formulas

Focal Distance (c) c  a 2  b2

c  ea

Distance From Center to Directrix (d) a d e

HYPERBOLA

HYPERBOLA

Key Formulas

Equation of Asymptotes y  k   m( x  h ) b m a

Transverse axis horizontal

a m b

Transverse axis vertical

REVIEW QUESTIONS 16. Find the eccentricity of the curve 9x2 – 4y2 -36x +8y = 4 a. 1.80 b. 1.70

c. 1.90 d. 1.60

REVIEW QUESTIONS 17. Find the length of the latus rectum of 9x2 – 4y2 -36x +8y = 4 a. 5 units b. 6 units

c. 9 units d. 8 units

REVIEW QUESTIONS 18. The distance between foci of a hyperbolas is 18 and the distance between directrices is 2. What is the eccentricity of the hyperbola?

a. 0.19 b. 3

c. 2 d. 1.6

REVIEW QUESTIONS 19. Find the eccentricity of a hyperbola whose transverse and conjugate axes are equal in length? a. 0.82 b. 1.41

c. 1.52 d. 1.73

REVIEW QUESTIONS 20. Find the equation of the asymptote of the hyperbola 4x2 – 9y2 = 36. a. x+3y=0 b. 3x+2y=0 c. 2x-3y=0 d. x-3y

y  k   m( x  h )

b m a

a m b

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