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    Analytic Geometry

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    Allen Kyle Sayo
    1. The document provides information on coordinate systems, lines, angles, distances, and conic sections including circles, parabolas, ellipses, and hyperbolas. 2. It also includes 30 problems related to these topics asking about distances, slopes, angles between lines, equations of lines and conic sections, and properties of geometric shapes. 3. The problems cover a wide range of skills including calculating distances and slopes, finding equations, determining properties of lines and conic sections, and applying geometric concepts.

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    Analytic Geometry

    Uploaded by

    Allen Kyle Sayo
    51 views65 pages
    1. The document provides information on coordinate systems, lines, angles, distances, and conic sections including circles, parabolas, ellipses, and hyperbolas. 2. It also includes 30 problems related to these topics asking about distances, slopes, angles between lines, equations of lines and conic sections, and properties of geometric shapes. 3. The problems cover a wide range of skills including calculating distances and slopes, finding equations, determining properties of lines and conic sections, and applying geometric concepts.

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    1. The document provides information on coordinate systems, lines, angles, distances, and conic sections including circles, parabolas, ellipses, and hyperbolas. 2. It also includes 30 problems related to these topics asking about distances, slopes, angles between lines, equations of lines and conic sections, and properties of geometric shapes. 3. The problems cover a wide range of skills including calculating distances and slopes, finding equations, determining properties of lines and conic sections, and applying geometric concepts.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    51 views65 pages

    Analytic Geometry

    Uploaded by

    Allen Kyle Sayo
    1. The document provides information on coordinate systems, lines, angles, distances, and conic sections including circles, parabolas, ellipses, and hyperbolas. 2. It also includes 30 problems related to these topics asking about distances, slopes, angles between lines, equations of lines and conic sections, and properties of geometric shapes. 3. The problems cover a wide range of skills including calculating distances and slopes, finding equations, determining properties of lines and conic sections, and applying geometric concepts.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 65

    Rectangular Coordinate System/Cartesian Plane:

    COORDNATE SYSTEM

    Coordinate(x, y)- n ordered pair used to


    designate position of a point in the coordinate
    plane. (e.g., the coordinate of the intersection of
    the x and y axes is called the origin (0,0))

    • Abscissa, x -the x-coordinate of a point


    • Ordinate, y -the y-ordinate of a point
    DISTANCE FORMULA
    SLOPE, m

    Slope of a Line-refers to the steepness of a line


    relative to the axes. Given two points,
    ANGLES AND SLOPES
    ANGLE BETWEEN LINES
    POINT TO A LINE
    PARALLEL LINES DISTANCE

    Distance between two parallel lines


    MIDPOINTS
    AREA OF TRIANGLE
    FORMS OF A LINE
    CONICS SECTIONS

    CONIC SECTIONS-are
    curves of the intersection
    between a plane and a
    vertical cone (‘nappe’).
    It is also the locus of all
    points which are at fixed
    ratio(e, eccentricity) from
    a given point (focus)
    CONICS SECTIONS
    CIRCLE
    PARABOLA
    PARABOLA
    PARABOLA
    ELLIPSE
    ELLIPSE

    -locus of points such that the sum of its distance


    from two fixed points (foci) is constant

    Focus (plu. Foci) – two fixed points where the


    length is held constant at 2a.
    ELLIPSE
    ELLIPSE
    ELLIPSE

    2b2
    LR 
    a
    HYPERBOLA
    HYPERBOLA
    HYPERBOLA
    Transverse axis-axis that passes through the two
    foci.

    Conjugate axis-axis perpendicular to the


    transverse axis.

    Auxiliary rectangle-the rectangle formed by the


    transverse and conjugate axes.
    HYPERBOLA
    HYPERBOLA

    2b 2
    LR 
    a
    HYPERBOLA
    HYPERBOLA
    PROBLEM 1

    Find the distance between the points (3, -5) and


    (-11, 20):

    A. 56.7
    B. 28.7
    C. 17.8
    D. 23.6
    PROBLEM 2

    Find the inclination of the line passing through (-


    5, 3) and (10, 7).
    A. 14.73
    B. 14.93
    C. 14.83
    D. 14.63
    PROBLEM 3

    Determine the coordinates of the point which


    is three-fifths of the way from the point (2, -5)
    to the point (-3, 5).
    A. (-1, 1)
    B. (-2, -1)
    C. (-1, -2)
    D. (1, -1)
    PROBLEM 4
    Find the angle between the lines 3x + 2y = 6 and
    x + y = 6.
    A. 12° 20’
    B. 11° 19’
    C. 14° 25’
    D. 13° 06’
    PROBLEM 5

    Find the angle formed by the lines

    2x + y – 8 = 0 and
    x + 3y + 4 = 0

    a. 30⁰ c. 45⁰
    b. 35⁰ d. 60⁰
    PROBLEM 6

    What is the distance of the line 3x + 4y


    = 5 to the point (4, -6)?
    a. 18/5 c. 11/3
    b. 17/5 d. 18/3
    PROBLEM 7

    What is the distance from a point (1, 3) to


    the line 4x + 3y + 12 = 0:

    a. 4 units c. 5 units
    b. 6 units d. 7 units
    PROBLEM 8

    Find the distance between the given lines 4x –


    3y = 12 and 4x – 3y = -8:

    a. 3 c. 4
    b. 5 d. 6
    PROBLEM 9
    The two points on the lines 2x = 3y + 4 = 0 which
    are at a distance 2 from the line 3x + 4y – 6 = 0
    are?
    A. (-5, 1) and (-5, 2)
    B. (64, -44) and (4, -4)
    C. (8, 8) and (12, 12)
    D. (44, -64) and (-4, 4)
    PROBLEM 10
    The segment from A(-1, 4) to B(2, -2) is
    extended by three times its own length at B. The
    terminal point is:
    a. (11, -24)
    b. (-11, -20)
    c. (11, -18)
    d. (11, -20)
    PROBLEM 11

    Given three vertices of a triangle whose


    coordinates are A(1,1), B(3, -3) and C(5, -3) find
    the area of the triangle
    a. 3
    b. 4
    c. 5
    d. 6
    PROBLEM 12

    If the points (-2, 3) and (-3, 5) lie on a straight


    line, then the equation of the line is:
    a. X – 2y = 1= 0
    b. 2x + y – 1= 0
    c. X + 2y – 1= 0
    d. 2x + y + 1 = 0
    PROBLEM 13
    The equation of a line that intercepts the x-axis
    at x = 4 and the y-axis at y = -6 is,
    A. 3x + 2y = 12
    B. 2x – 3y = 12
    C. 3x – 2y = 12
    D. 2x – 3y = 12
    PROBLEM 14
    Determine the equation of the line
    perpendicular to 7x -8y + 12 = 0 and intersects
    the point (1/4, 3):
    a. 8x + 7y – 23= 0 c. 9x + 7y – 21= 0
    b. 8x - 7y + 23= 0 d. 9x - 7y + 21= 0
    PROBLEM 15
    Find an equation for the line that is parallel to
    the line 5x + 3y = 4 and x-intercept = 5
    a. 5x + 3y = 15
    b. 3x - 5y = 15
    c. 5x + 3y = 25
    d. 3x - 5y = -25
    PROBLEM 16
    Find the equation of the line through point (3, 1)
    and is perpendicular to the line x + 5y +5 = .
    A. 5x – 2y = 14
    B. 5x – y = 14
    C. 2x – 5y = 14
    D. 2x + 5y = 14
    PROBLEM 17
    The equation of the line through (-3, -5) parallel
    to 7x + 2y – 4 = 0 is

    A. 7x + 2y + 31 = 0
    B. 7x – 2y + 30 = 0
    C. 7x – 2y – 4 = 0
    D. 2x + 7y + 30 = 0
    PROBLEM 18
    Determine the center and radius, respectively, of the
    circle:

    x 2  y 2  6x  14y  33  0
    A. (3, 14) and 4
    B. (-3, 7) and 5
    C. (3, 7) and 5
    D. (-3, 14) and 4
    PROBLEM 19
    The equation of the circle with center at (-2, 3)
    and which is tangent to the line 20x – 21y – 42 =
    0.

    A. x2 + y2 + 4x – 6y – 12 = 0
    B. x2 + y2 + 4x – 6y + 12 = 0
    C. x2 + y2 + 4x + 6y – 12 = 0
    D. x2 + y2 – 4x – 6y – 12 = 0
    PROBLEM 20
    A circle has a diameter whose ends are at (-3, 2)
    and (12, -6). Its Equation is:

    A. 4x2 + 4y2 – 36x + 16y + 192 = 0


    B. 4x2 + 4y2 – 36x + 16y – 192 = 0
    C. 4x2 + 4y2 – 36x – 16y – 192 = 0
    D. 4x2 + 4y2 – 36x – 16y + 192 = 0
    PROBLEM 18
    A parabolic arch is 112 ft high at the center. The
    bottom has a width of 247 ft. what is the height
    of the arch if one stands 10 ft from one end?

    A. 16.7 ft
    B. 17.4 ft
    C. 10.9 ft
    D. 8.5 ft
    PROBLEM 16
    The support of a suspension bridge is parabolic such
    that the supports has a height of 320 ft. The bridge is
    1800 ft long and the lowest point of the cable is 50 m
    from the bridge. Find the height of the cable if the a car
    is 500 ft from one end?

    A. 60 m B. 120 m C. 170 m D. 103 m


    PROBLEM 17
    Supposing that a transmission cable sags in a parabolic
    curve between two posts each 45 ft high. When one
    stands 7 ft from the center, the height of the cable is 21 ft
    and when on stands 9 ft from the center, the height of
    the cable is 25 ft Determine:

    The distance between two poles:


    A. 43’ B. 31’ C. 36’ D. 40’

    The distance of the sag from the ground:


    A. 13.7’ B. 12.9’ C. 11.4’ D. 14.9’
    PROBLEM 18
    Find the equation of the ellipse that has
    vertices at (0 , ± 10) and has eccentricity of
    0.8.

    A. x^2/36 + y^2/100 = 1
    B. x^2/9 + y^2/25 = 1
    C. x^2/36 + y^2/25 = 1
    D. x^2/9 + y^2/36 = 1
    PROBLEM 19
    An elliptical arch has a span of 287 ft and a
    height of 78 ft. Determine the distance from one
    end where the height of the arch is 50ft:

    A. 42.7 ft
    B. 33.5 f
    C. 67.6 ft
    D. 53.8ft
    PROBLEM 20
    A cable sags such that the supports are 42 ft poles. The lowest
    point of the cable is 12 ft from the ground. If a surveyor situates
    the transit 20 ft from the center, the cable is 15 ft high. Find the
    distance between the two poles.

    A. 89 ft B. 56 ft C. 113 ft D. 74 m
    PROBLEM 21
    An arch in the form of a semi-ellipse is 52 ft
    wide at the base and has a height of 20 ft. How
    wide is the arch at a height of 12 ft above the
    base?
    a. 41.6 ft b. 22.5 ft
    c. 35.5 ft d. 17.7 ft
    PROBLEM 22
    A transmission cable sags at the center such that
    the curve is a catenary(hyperbolic) between the 15
    m high poles, 19 m apart and the lowest point is 9
    m from the ground. Determine the height of the
    cable if on stands 5 m from the center:

    A. 10 m
    B. 11 m
    C. 12 m
    D. 13 m
    PROBLEM 23
    An elliptical arch is 500 meters wide. The
    maximum height of the arch is unknown.
    However, it is known that when one stands 50
    meters from one end, the height is 200 m. The
    maximum height of the arch is:

    A. 667 m B. 333 m. C. 450 m D. 233 m


    PROBLEM 24
    Find the equation of the hyperbola with vertices
    at (0 , ± 6) and eccentricity of 5 / 3.

    A. y^2/36 – x^2/64 = 1
    B. y^2/4 – x^2/64 = 1
    C. y^2/25 – x^2/64 = 1
    D. y^2/25 – x^2/4 = 1
    PROBLEM 25
    For a conics, the V(±3, 0) and F(±7, 0). Find the
    equation of the conics:

    A. x^2/9 + y^2/49 = 1
    B. x^2/9 – y^2/40 = 1
    C. x^2/40 – y^2/49 = 1
    D. x^2/49+ y^2/9 = 1
    PROBLEM 26
    A bridge is built in the shape of a semielliptical
    arch. It has a span of 112 feet. The height of the
    arch 28 feet from the center is to be 6 feet. Find
    the height of the arch at its center.
    a. 6.93 ft b. 6.2 ft
    c. 28.16 ft d. 12 ft
    PROBLEM 27
    A hyperbolic cable has a span of 220 feet and
    between two poles each 40 feet high. The
    lowest point of the cable is 20 feet from the
    ground, the height of the cable 50 meters from
    one pole is:
    a. 27.5 ft b. 16.5 ft
    b. 32.5 ft d. 18.5 ft
    PROBLEM 28
    A parabolic arch of a bridge sags 45 meters at a
    span of L meters between two locations. The
    lowest point at the of the sag at the center is 5
    meters from the ground. If one moves
    horizontally 100 meters from the center, the
    height of the cable is 15 meters. What is the
    span L of the bridge?
    a. 573 m b. 780 c. 425 d. 400
    PROBLEM 29
    An arch 18 m high has the form of parabola with
    a vertical axis. The length of a horizontal beam
    placed across the arch 8m from the top is 64m.
    Find the width at the bottom.
    a. 86m b. 96m c. 106m d. 76m
    PROBLEM 30
    A hyperbolic cable hangs such that the sag is 45
    ft from the level ground. If one stands 50 ft from
    the center. The cable is 134 m high. If the poles
    are 200 ft apart, the height of the poles is
    nearly:
    a. 256 m c. 341 m
    b. 189 m d. 201 m

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