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CIVIL ENGINEERING BOARD EXAMS PROBLEMS PHILIPPINES - October 26, 2020
CONIC SECTIONS
CONIC SECTIONS - In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three
types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a
fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their
properties.
Types of Conic Sections
- Parabolas
- Ellipses
- Hyperbolas
PARABOLAS – set of all points in a plane that are equidistant from a fixed line (known as DIRECTRIX) and a fixed point (known as the FOCUS) that is not
on the line.
VERTEX – The uppermost , lowermost , leftmost or rightmost point of the parabola
AXIS OF SYMMETRY – line passing through the focus and perpendicular to the directrix.
LATUS RECTUM – chord passing through the focus and parallel to the directrix or perpendicular to axis.
GENERAL EQUATION OF A PARABOLA:
GENERAL FORM: STANDARD FORM:
Conditions in solving the parabola:
- Three points along the parabola and an axis (either vertical or horizontal) - General Form
- Vertex (h,k) , distance from the vertex to focus a and axis – Standard Form
- Vertex (h,k) and location of focus – Standard Form
ECCENTRICITY – ratio of its distance from the focus and from the directrix. Parabola’s eccentricity is equal to 1.
EXAMPLES:
- Find the vertex of the parabola x = y
2
SOLUTION:
- Write the equation of the directrix of the parabola y
2
SOLUTION:
ELLIPSES
An ellipse is the set of all points in a plane whose distances of two fixed points (foci) is constant. The midpoint of the segment connecting the foci
is the center of the ellipse.
EQUATIONS OF THE ELLIPSE
STANDARD FORM GENERAL FORM
ELEMENTS OF ELLIPSE:
Conditions in Solving an Ellipse:
- Four points along the ellipse – general form
- Center (h,k) , semi-major axis a and semi-minor axis b – standard form
Solving for the center of the ellipse in general form:
EXAMPLES:
- Find the center of this ellipse:
2
2
SOLUTION:
- Find the eccentricity of the ellipse in the figure shown.
SOLUTION:
- Of the nine planetary orbits in our solar system, Pluto’s has the greatest eccentricity, 0.248. Astronomers have determined that the orbit is about
29.646 AU (astronomical units) from the sun at its closest point to the sun (perihelion). The length of the semi-major axis is about 39.482 AU. 1
AU is the average distance between the sun and Earth, about 9.3 10 ^7 miles. Find the distance of Pluto from the sun at its farthest point.
SOLUTION:
ELEMENTS OF HYPERBOLA EQUATIONS OF ASYMPTOTES
EXAMPLES:
- Find the y – intercepts of the hyperbola 4 y
2
2
SOLUTION:
- What is the length of the latus rectum of the hyperbola x
2
/25 – y
2
/64 = 1?
SOLUTION:
- The design layout of a cooling tower is shown in the figure. The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their
closest, the sides of the tower are 60 meters apart. Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the
center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final
values to four decimal places.
SOLUTION:
EXERCISES – Answer the following questions.
- Identify the foci of the hyperbola y
2
/49 – x
2
/32 = 1. Ans. (0,9)
- Label the asymptote of the hyperbola 9x
2
2
- 36x – 40y - 388 = 0. Ans. y = ±3/2 (x – 2 ) – 5
- Locate the center of the hyperbola (y + 4)
2
/36 - (x – 2)
2
/25 = 1. Ans. (2, - 4)
- Stations A and B are 150 kilometers apart and send a simultaneous radio signal to the ship. The signal from B arrives 0.0003 seconds before the
signal from A. If the signal travels 300,000 kilometers per second, find the equation of the hyperbola on which the ship is positioned.
Ans. x
2
/2025 – y
2
- The length of the latus rectum of a hyperbola is equal to 1 8 and the distance between the foci is 12. Compute the equation of the asymptote of the
hyperbola. Ans. y = √ 3 x
- Determine the vertices of the hyperbola (x + 7)
2
/36 – (y + 4)
2
/64 = 1. Ans. ( - 13, - 4) and ( - 1 , - 4)
- Find the equation of the hyperbola with the distance between foci having equal to 18 and between directrices having equal to 2. Ans.
8x
2
2
- Locate the foci of the hyperbola (x + 2)
2
/9 – (y – 5)
2
/ 4 9 = 1. Ans. (-2 , ± √58)
- (CE Board) The equilateral hyperbola xy = 8 and has the x-axis and y-axis as asymptote. Compute the eccentricity of the hyperbola. Ans. 1.
- Cross section of a Nuclear cooling tower is in the shape of a hyperbola with equation (x^2/302) - (y^2/442) = 1. The tower is 150 m tall and the
distance from the top of the tower to the centre of the hyperbola is half the distance from the base of the tower to the centre of the hyperbola. Find
the diameter of the top and base of the tower. Ans. 45.41 and 74.45 m
NEXT TOPICS ON October 30, 2020:
- Determination of Conics, Tangent and Normal to Conics
- Rotation and Translation of Axes
- Parametric Equations
REFERENCES:
- Introductory and Intermediate Algebra by Blitzer
- Engineering Mathematics by Gillesania
- https://en.wikipedia.org/wiki/Conic_section
- https://www.onlinemath4all.com/finding-the-vertex-focus-directrix-and-latus-rectum-of-the-parabola.html
- https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Quadratic-relations-and-conic-
sections.faq.question.856050.html
- Algebra and Trigonometry by Barnett
- Glencoe’s Advanced Mathematical Concepts
- https://www.easyteacherworksheets.com/pages/pdf/math/geometry/parabolas/54.html
- Engineering Mathematics by Besavilla
- Intermediate Algebra by Elayn Martin
- Analytic Geometry by Rainville
- https://mi01000971.schoolwires.net/cms/lib/MI01000971/Centricity/Domain/433/Parabolas%20WS%20D1.pdf
- Schaum’s Outlines of College Mathematics
- https://www.purplemath.com/modules/parabola3.htm
- https://www.mathwarehouse.com/ellipse/eccentricity-of-ellipse.php#problem
- https://www.cbsd.org/cms/lib/PA01916442/Centricity/Domain/2023/333202_1003_744-752.pdf
- Algebra and Trigonometry by Openstax
- https://www.emathzone.com/tutorials/geometry/applications-of-ellipse.html
- Algebra and Trigonometry by Cynthia Young
- https://www.mathopolis.com/questions/q.html?id=838&t=mif&qs=835_3336_836_3337_837_3338_838_3339_9068_9069&site=1&ref=2f67656f
6d657472792f6879706572626f6c612e68746d6c&title=4879706572626f6c
- http://www.opentextbookstore.com/precalc/2/Precalc9-2.pdf
- Precalculus Demystified
- Analytic Geometry by Rainville
- http://www.phschool.com/atschool/ap_misc/dwfk_precalc/pdfs/8e/Ch8_Section3.pdf
