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2.2 Trigonometry and Analytical Geometry
PRC Civil Engineer Licensure Examination CELE
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1 Chapter 1: Examination Overview, RA 544 & Professional Practice
1.1 CELE Structure, Format, Scoring, and 2026 Subject Order 1.2 Republic Act No. 544: The Civil Engineering Law and Board Regulations 1.3 Civil Engineering Ethics, Contracts, and Professional Practice
2 Chapter 2: Engineering Mathematics, Numerical Methods & Physics
2.1 Advanced Algebra, Progressions & Word Problems 2.2 Trigonometry and Analytical Geometry 2.3 Differential and Integral Calculus 2.4 Differential Equations and Numerical Methods 2.5 Series, Hyperbolic Functions, Multivariable Calculus, and Laplace Transforms 2.6 Numerical Methods: Linear Systems, Interpolation, Curve Fitting, and ODE Solvers 2.7 Physics for Engineers: Kinematics, Newton's Laws, Work-Energy, and Momentum
3 Chapter 3: Engineering Data Analysis & Engineering Economy
3.1 Engineering Data Analysis and Statistical Applications 3.2 Parameter Estimation, Linear Regression, Correlation, and Queuing Models 3.3 Engineering Economy: Time Value of Money and Interest Formulas 3.4 Project Evaluation, Depreciation, and Benefit-Cost Analysis
4 Chapter 4: Surveying Fundamentals, Construction Layout & Route Surveying
4.1 Errors, Corrections, and Pacing/Taping 4.2 Leveling, Earthworks, and Mass Haul Diagrams 4.3 Traverse Computations, Balancing, and Area Determinations 4.4 Route Surveying: Horizontal and Vertical Curves
5 Chapter 5: Highway & Transportation Engineering
5.1 Highway Geometric Design and Sight Distances 5.2 Traffic Engineering, Flow Fundamentals, and Pavement Design 5.3 Highway Policy, Route Location, Drainage, and Earthwork Slopes 5.4 Transportation Planning, Network Analysis, and Intermodal Systems
6 Chapter 6: Construction Materials, Methods, Management & Safety
6.1 Materials for Construction: Concrete, Formwork, Masonry, Steel, and Wood Products 6.2 Construction Equipment, Formwork, and Construction Methods 6.3 Construction Occupational Safety and Health: RA 11058 and DOLE DO 13 6.4 Construction Project Management, CPM/PERT, and Resource Scheduling 6.5 Construction Principles: Design Requirements, Project Organization, Cost Control, and Value Engineering 6.6 Quantity Surveying, Unit Cost Analysis, and Project Cost Estimates 6.7 Construction Trades and Specialties: Plumbing, Plastics, Doors and Windows, Finishes, and Electrical
7 Chapter 7: Fluid Properties, Hydrostatics & Buoyancy
7.1 Fluid Properties, Units, and Pressure Measurement 7.2 Hydrostatic Pressure on Plane and Curved Surfaces 7.3 Buoyancy, Flotation, and Accelerated Fluid Masses
8 Chapter 8: Hydrodynamics, Pipe Flow, Open Channels & Water Supply
8.1 Fluid Dynamics: Continuity and Energy Equations 8.2 Flow Measurement: Orifices, Weirs, and Venturi Meters 8.3 Fluid Flow in Pipes, Friction Losses, and Pipe Networks 8.4 Open Channel Flow, Manning's Equation, and Hydraulic Jump 8.5 Water Supply: Demand, Storage, Pumping, Distribution, and Treatment
9 Chapter 9: Soil Properties, Seepage & Stresses in Soil Mass
9.1 Soil Composition, Weight-Volume Phase Relationships 9.2 Soil Classification Systems (USCS & AASHTO) and Atterberg Limits 9.3 Soil Permeability, Seepage, Flow Nets, and Effective Stress 9.4 Stresses in Soil Mass, Stress Distribution, and Immediate and Secondary Settlement
10 Chapter 10: Consolidation, Soil Strength, Earth Pressures & Foundations
10.1 Soil Compaction and Soil Compressibility/Consolidation 10.2 Shear Strength of Soils and Mohr-Coulomb Failure Criteria 10.3 Lateral Earth Pressures (Rankine & Coulomb) and Retaining Walls 10.4 Shallow and Deep Foundation Bearing Capacity and Slope Stability
11 Chapter 11: Engineering Mechanics & Statics
11.1 Force Systems, Resultants, and Equilibrium of Rigid Bodies 11.2 Centroids, Moments of Inertia, and Friction 11.3 Cables, Trusses, and Internal Forces in Members
12 Chapter 12: Strength of Materials
12.1 Axial Stress, Strain, and Deformation 12.2 Torsion, Flanged Bolt Couplings, and Thin-Walled Pressure Vessels 12.3 Beam Stresses: Flexural and Transverse Shear 12.4 Beam Deflections, Combined Stresses, and Mohr's Circle
13 Chapter 13: Structural Analysis & NSCP Load Criteria
13.1 Indeterminate Structures: Approximate and Classical Methods 13.2 Moment Distribution Method and Slope Deflection 13.3 Influence Lines for Determinate and Indeterminate Structures 13.4 NSCP Minimum Design Loads: Dead, Live, Wind, and Earthquake
14 Chapter 14: Reinforced Concrete Design (NSCP 2015 / USD)
14.1 Ultimate Strength Design (USD) Principles and Singly Reinforced Beams 14.2 Doubly Reinforced Beams and T-Beams 14.3 Shear and Diagonal Tension, Stirrup Design, and Development Length 14.4 Reinforced Concrete Columns and Isolated Footing Design
15 Chapter 15: Steel, Timber & Prestressed Design and Materials Testing
15.1 Structural Steel Tension Members and Connections 15.2 Structural Steel Compression Members and Beam Design 15.3 Timber Design Principles, Working Stress, and Sawn Lumber Beams 15.4 Prestressed Concrete Fundamentals and Prestress Losses 15.5 Construction Materials Testing and Concrete Mix Proportioning
2.2 Trigonometry and Analytical Geometry
Key Takeaways Explain This Topic AI
The Laws of Sines and Cosines resolve oblique planar triangles, with the ambiguous case (SSA) evaluated by comparing side length a to altitude h = b sin A.
Spherical trigonometry governs geodetic and terrestrial coordinate calculations; right spherical triangles are solved using Napier's Rules with co-parts for hypotenuse and angles.
The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 is classified via discriminant B² - 4AC, with conic eccentricity e = c/a defining circle, ellipse, parabola, and hyperbola.
Perpendicular distance from point (x₁, y₁) to line Ax + By + C = 0 is d = |Ax₁ + By₁ + C| / √(A² + B²), while perpendicular lines satisfy m₁ · m₂ = -1.
Last updated: October 2026
2.2 Trigonometry and Analytical Geometry
Trigonometry and analytical geometry form the analytical foundation for civil engineering surveying, structural frame geometry, geodetic positioning, and transportation highway alignment. Board exam questions in this domain evaluate candidates on planar triangle mechanics, spherical geodesy using Napier's rules, straight-line analytical relationships, and the canonical formulations of conic sections (circles, parabolas, ellipses, and hyperbolas).
- Planar Trigonometry & Oblique Triangle Solutions
- Fundamental Identities
Trigonometric problem-solving relies on instant recognition of core identities:
Pythagorean Identities: sin 2 θ + cos 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + \tan^2\theta = \sec^2\theta 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ 1 + \cot^2\theta = \csc^2\theta 1 + cot 2 θ = csc 2 θ
Double-Angle Identities: sin 2 θ = 2 sin θ cos θ \sin 2\theta = 2\sin\theta\cos\theta sin 2 θ = 2 sin θ cos θ cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta cos 2 θ = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2 sin 2 θ tan 2 θ = 2 tan θ 1 − tan 2 θ \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta} tan 2 θ = 1 − t a n 2 θ 2 t a n θ
Half-Angle Identities: sin θ 2 = ± 1 − cos θ 2 , cos θ 2 = ± 1 + cos θ 2 , tan θ 2 = 1 − cos θ sin θ = sin θ 1 + cos θ \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}, \quad \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}, \quad \tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta} sin 2 θ = ± 2 1 − c o s θ , cos 2 θ = ± 2 1 + c o s θ , tan 2 θ = s i n θ 1 − c o s θ = 1 + c o s θ s i n θ
Oblique Triangle Laws
Any non-right (oblique) planar triangle with interior angles A , B , C A, B, C A , B , C and opposing sides a , b , c a, b, c a , b , c is solved via two fundamental laws:
Law of Sines: a sin A = b sin B = c sin C = 2 R \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R s i n A a = s i n B b = s i n C c = 2 R Where R R R is the radius of the triangle's circumscribed circle ( circumradius ).
Law of Cosines: a 2 = b 2 + c 2 − 2 b c cos A a^2 = b^2 + c^2 - 2bc\cos A a 2 = b 2 + c 2 − 2 b c cos A b 2 = a 2 + c 2 − 2 a c cos B b^2 = a^2 + c^2 - 2ac\cos B b 2 = a 2 + c 2 − 2 a c cos B c 2 = a 2 + b 2 − 2 a b cos C c^2 = a^2 + b^2 - 2ab\cos C c 2 = a 2 + b 2 − 2 ab cos C (Preferred when two sides and the included angle (SAS) or all three sides (SSS) are known).
The Ambiguous Case (Side-Side-Angle / SSA)
When given sides a , b a, b a , b and acute angle A A A ( A < 90 ∘ A < 90^\circ A < 9 0 ∘ ), calculate the altitude h = b sin A h = b\sin A h = b sin A :
If a < h a < h a < h : No triangle exists (side a a a is too short to close the figure).
If a = h a = h a = h : Exactly one right triangle exists ( B = 90 ∘ B = 90^\circ B = 9 0 ∘ ).
If h < a < b h < a < b h < a < b : Two distinct triangles exist (one acute where B 1 < 90 ∘ B_1 < 90^\circ B 1 < 9 0 ∘ , one obtuse where B 2 = 180 ∘ − B 1 B_2 = 180^\circ - B_1 B 2 = 18 0 ∘ − B 1 ).
If a ≥ b a \ge b a ≥ b : Exactly one triangle exists .
- Planar Triangle Area Formulations
- Base and Altitude: K = 1 2 b h K = \frac{1}{2} b h K = 2 1 bh
Two Sides and Included Angle: K = 1 2 a b sin C = 1 2 b c sin A = 1 2 a c sin B K = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B K = 2 1 ab sin C = 2 1 b c sin A = 2 1 a c sin B
Heron's Formula (Three Sides Known): K = s ( s − a ) ( s − b ) ( s − c ) , where semi-perimeter s = a + b + c 2 K = \sqrt{s(s-a)(s-b)(s-c)}, \quad \text{where semi-perimeter } s = \frac{a + b + c}{2} K = s ( s − a ) ( s − b ) ( s − c ) , where semi-perimeter s = 2 a + b + c
Inscribed Circle (Inradius r r r ): K = r ⋅ s ⟹ r = K s K = r \cdot s \implies r = \frac{K}{s} K = r ⋅ s ⟹ r = s K
Circumscribed Circle (Circumradius R R R ): K = a b c 4 R ⟹ R = a b c 4 K K = \frac{abc}{4R} \implies R = \frac{abc}{4K} K = 4 R ab c ⟹ R = 4 K ab c
Spherical Trigonometry in Geodetic Civil Engineering
On the Earth's surface (modeled as a sphere of radius R ≈ 6 , 371 km R \approx 6,371\text{ km} R ≈ 6 , 371 km ), spherical triangles are formed by the intersections of three great circles . The lengths of sides a , b , c a, b, c a , b , c are subtended central angles measured in degrees or radians.
- Fundamental Spherical Properties
- Sum of sides: 0 ∘ < a + b + c < 360 ∘ 0^\circ < a + b + c < 360^\circ 0 ∘ < a + b + c < 36 0 ∘
Sum of interior angles: 180 ∘ < A + B + C < 540 ∘ 180^\circ < A + B + C < 540^\circ 18 0 ∘ < A + B + C < 54 0 ∘
Spherical Excess ( E E E ): The angular amount by which the sum of interior angles exceeds planar Euclidean geometry: E = A + B + C − 180 ∘ E = A + B + C - 180^\circ E = A + B + C − 18 0 ∘
Surface Area of a Spherical Triangle: Area = π R 2 E 180 ∘ = R 2 E radians \text{Area} = \frac{\pi R^2 E}{180^\circ} = R^2 E_{\text{radians}} Area = 18 0 ∘ π R 2 E = R 2 E radians
Napier's Rules for Right Spherical Triangles ( C = 90 ∘ C = 90^\circ C = 9 0 ∘ )
In a right spherical triangle where angle C = 90 ∘ C = 90^\circ C = 9 0 ∘ , the five remaining parts are arranged circularly in five sequential sectors: a , b , co- A = ( 90 ∘ − A ) , co- c = ( 90 ∘ − c ) , co- B = ( 90 ∘ − B ) a, \quad b, \quad \text{co-}A = (90^\circ - A), \quad \text{co-}c = (90^\circ - c), \quad \text{co-}B = (90^\circ - B) a , b , co- A = ( 9 0 ∘ − A ) , co- c = ( 9 0 ∘ − c ) , co- B = ( 9 0 ∘ − B )
Napier established two universal mnemonic rules:
Rule 1 (Tan-Ad): The sine of any middle part equals the product of the tangents of its two adjacent parts: sin ( middle ) = tan ( adjacent 1 ) ⋅ tan ( adjacent 2 ) \sin(\text{middle}) = \tan(\text{adjacent}_1) \cdot \tan(\text{adjacent}_2) sin ( middle ) = tan ( adjacent 1 ) ⋅ tan ( adjacent 2 )
Rule 2 (Cos-Op): The sine of any middle part equals the product of the cosines of its two opposite parts: sin ( middle ) = cos ( opposite 1 ) ⋅ cos ( opposite 2 ) \sin(\text{middle}) = \cos(\text{opposite}_1) \cdot \cos(\text{opposite}_2) sin ( middle ) = cos ( opposite 1 ) ⋅ cos ( opposite 2 )
(Example Application: Choosing co- c \text{co-}c co- c as the middle part gives sin ( co- c ) = cos ( a ) cos ( b ) ⟹ cos c = cos a cos b \sin(\text{co-}c) = \cos(a)\cos(b) \implies \cos c = \cos a \cos b sin ( co- c ) = cos ( a ) cos ( b ) ⟹ cos c = cos a cos b .)
Analytical Geometry of Straight Lines
In the 2D Cartesian plane, linear alignments govern road centerlines, pipeline easements, and structural grids:
Geometric Parameter Mathematical Formula Notes
Distance Between Points d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Euclidean metric
Slope of Line m = y 2 − y 1 x 2 − x 1 = tan θ m = \frac{y_2 - y_1}{x_2 - x_1} = \tan\theta m = x 2 − x 1 y 2 − y 1 = tan θ Undefined for vertical lines
Angle Between Lines tan ϕ = ∣ m 2 − m 1 1 + m 1 m 2 ∣ \tan\phi = \left\lvert\frac{m_2 - m_1}{1 + m_1 m_2}\right\rvert tan ϕ = 1 + m 1 m 2 m 2 − m 1 Acute angle between lines
Parallel Lines Condition m 1 = m 2 m_1 = m_2 m 1 = m 2 Equal inclinations
Perpendicular Lines Condition m 1 ⋅ m 2 = − 1 ⟺ m 2 = − 1 m 1 m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1} m 1 ⋅ m 2 = − 1 ⟺ m 2 = − m 1 1 Negative reciprocal slopes
Point-to-Line Distance d = ∣ A x 1 + B y 1 + C ∣ A 2 + B 2 d = \frac{\lvert Ax_1 + By_1 + C\rvert}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ A x 1 + B y 1 + C ∣ Shortest normal offset from ( x 1 , y 1 ) (x_1, y_1) ( x 1 , y 1 ) to A x + B y + C = 0 Ax + By + C = 0 A x + B y + C = 0
Distance Between Parallel Lines d = ∣ C 1 − C 2 ∣ A 2 + B 2 d = \frac{\lvert C_1 - C_2\rvert}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ C 1 − C 2 ∣ For A x + B y + C 1 = 0 Ax + By + C_1 = 0 A x + B y + C 1 = 0 and A x + B y + C 2 = 0 Ax + By + C_2 = 0 A x + B y + C 2 = 0
Conic Sections: General & Canonical Equations
The general second-degree Cartesian equation represents all conic sections: A x 2 + B x y + C y 2 + D x + E y + F = 0 Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 A x 2 + B x y + C y 2 + D x + E y + F = 0
Identification via the Conic Discriminant ( B 2 − 4 A C B^2 - 4AC B 2 − 4 A C )
When axes are unrotated ( B = 0 B = 0 B = 0 ):
Parabola ( B 2 − 4 A C = 0 B^2 - 4AC = 0 B 2 − 4 A C = 0 ): Either A = 0 A = 0 A = 0 or C = 0 C = 0 C = 0 (only one squared term).
Ellipse ( B 2 − 4 A C < 0 B^2 - 4AC < 0 B 2 − 4 A C < 0 ): A A A and C C C have the same algebraic sign ( A ⋅ C > 0 A \cdot C > 0 A ⋅ C > 0 ).
Circle: Special case of ellipse where A = C A = C A = C and B = 0 B = 0 B = 0 .
Hyperbola ( B 2 − 4 A C > 0 B^2 - 4AC > 0 B 2 − 4 A C > 0 ): A A A and C C C have opposite algebraic signs ( A ⋅ C < 0 A \cdot C < 0 A ⋅ C < 0 ).
Equilateral / Rectangular Hyperbola: Occurs when A = − C A = -C A = − C .
Classification by Eccentricity ( e e e )
Eccentricity is defined as the fixed ratio of the distance from any point on the curve to the focus ( d F d_F d F ) over its distance to the directrix line ( d D d_D d D ): e = d F / d D e = d_F / d_D e = d F / d D .
- e = 0 e = 0 e = 0 : Circle
- 0 < e < 1 0 < e < 1 0 < e < 1 : Ellipse ( e = c / a e = c/a e = c / a )
- e = 1 e = 1 e = 1 : Parabola ( e = 1 e = 1 e = 1 )
- e > 1 e > 1 e > 1 : Hyperbola ( e = c / a e = c/a e = c / a )
- Canonical Geometries of Conic Sections
- 1. The Circle
Standard center-radius form with center at ( h , k ) (h, k) ( h , k ) and radius r r r : ( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2 General form: x 2 + y 2 + D x + E y + F = 0 x^2 + y^2 + Dx + Ey + F = 0 x 2 + y 2 + D x + E y + F = 0 , where h = − D / 2 h = -D/2 h = − D /2 , k = − E / 2 k = -E/2 k = − E /2 , and r = h 2 + k 2 − F r = \sqrt{h^2 + k^2 - F} r = h 2 + k 2 − F .
2. The Parabola
A parabola is the locus of points equidistant from the focus and the directrix line ( e = 1 e = 1 e = 1 ). The parameter p p p is the directed focal distance from vertex to focus.
Vertical Axis of Symmetry (Opens Up/Down): ( x − h ) 2 = 4 p ( y − k ) (x - h)^2 = 4p(y - k) ( x − h ) 2 = 4 p ( y − k )
- Vertex: ( h , k ) (h, k) ( h , k )
- Focus: ( h , k + p ) (h, k + p) ( h , k + p )
- Directrix: y = k − p y = k - p y = k − p
- Length of Latus Rectum: L R = ∣ 4 p ∣ LR = |4p| L R = ∣4 p ∣
Horizontal Axis of Symmetry (Opens Left/Right): ( y − k ) 2 = 4 p ( x − h ) (y - k)^2 = 4p(x - h) ( y − k ) 2 = 4 p ( x − h )
- Vertex: ( h , k ) (h, k) ( h , k )
- Focus: ( h + p , k ) (h + p, k) ( h + p , k )
- Directrix: x = h − p x = h - p x = h − p
- Length of Latus Rectum: L R = ∣ 4 p ∣ LR = |4p| L R = ∣4 p ∣
- 3. The Ellipse
The locus of points such that the sum of distances to two fixed foci is constant ( d 1 + d 2 = 2 a d_1 + d_2 = 2a d 1 + d 2 = 2 a ). Here, a a a is the semi-major axis, b b b is the semi-minor axis, and c c c is the focal distance, governed by the Pythagorean relation: a 2 = b 2 + c 2 ⟹ c = a 2 − b 2 a^2 = b^2 + c^2 \implies c = \sqrt{a^2 - b^2} a 2 = b 2 + c 2 ⟹ c = a 2 − b 2
Horizontal Major Axis: ( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 ( a > b ) \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a > b) a 2 ( x − h ) 2 + b 2 ( y − k ) 2 = 1 ( a > b )
- Foci: ( h ± c , k ) (h \pm c, k) ( h ± c , k )
- Directrices: x = h ± a e = h ± a 2 c x = h \pm \frac{a}{e} = h \pm \frac{a^2}{c} x = h ± e a = h ± c a 2
- Length of Latus Rectum: L R = 2 b 2 a LR = \frac{2b^2}{a} L R = a 2 b 2
Vertical Major Axis: ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 ( a > b ) \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (a > b) b 2 ( x − h ) 2 + a 2 ( y − k ) 2 = 1 ( a > b )
- Foci: ( h , k ± c ) (h, k \pm c) ( h , k ± c )
- Directrices: y = k ± a e = k ± a 2 c y = k \pm \frac{a}{e} = k \pm \frac{a^2}{c} y = k ± e a = k ± c a 2
- 4. The Hyperbola
The locus of points such that the absolute difference of distances to two foci is constant ( ∣ d 1 − d 2 ∣ = 2 a |d_1 - d_2| = 2a ∣ d 1 − d 2 ∣ = 2 a ). Here, a a a is the semi-transverse axis, b b b is the semi-conjugate axis, and c c c is the focal distance: c 2 = a 2 + b 2 ⟹ c = a 2 + b 2 c^2 = a^2 + b^2 \implies c = \sqrt{a^2 + b^2} c 2 = a 2 + b 2 ⟹ c = a 2 + b 2
Horizontal Transverse Axis: ( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1
- Foci: ( h ± c , k ) (h \pm c, k) ( h ± c , k )
- Asymptotes: y − k = ± b a ( x − h ) y - k = \pm \frac{b}{a}(x - h) y − k = ± a b ( x − h )
- Length of Latus Rectum: L R = 2 b 2 a LR = \frac{2b^2}{a} L R = a 2 b 2
- Directrices: x = h ± a e = h ± a 2 c x = h \pm \frac{a}{e} = h \pm \frac{a^2}{c} x = h ± e a = h ± c a 2
Vertical Transverse Axis: ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 a 2 ( y − k ) 2 − b 2 ( x − h ) 2 = 1
- Asymptotes: y − k = ± a b ( x − h ) y - k = \pm \frac{a}{b}(x - h) y − k = ± b a ( x − h )
- Polar Coordinates & Conversions
The relationship between Cartesian coordinates ( x , y ) (x, y) ( x , y ) and polar coordinates ( r , θ ) (r, \theta) ( r , θ ) is defined by: x = r cos θ , y = r sin θ x = r\cos\theta, \quad y = r\sin\theta x = r cos θ , y = r sin θ r 2 = x 2 + y 2 , tan θ = y x r^2 = x^2 + y^2, \quad \tan\theta = \frac{y}{x} r 2 = x 2 + y 2 , tan θ = x y
Conic Sections in Polar Form (Focus at Pole): r = e ⋅ d 1 ± e cos θ or r = e ⋅ d 1 ± e sin θ r = \frac{e \cdot d}{1 \pm e\cos\theta} \quad \text{or} \quad r = \frac{e \cdot d}{1 \pm e\sin\theta} r = 1 ± e c o s θ e ⋅ d or r = 1 ± e s i n θ e ⋅ d Where e e e is eccentricity and d d d is the distance from the focus (pole) to the directrix.
- Step-by-Step Worked Coordinate Geometry Problem
- Worked Example: Siting a Highway Tangent and Retention Basin
Problem: A civil engineering surveyor determines that the centerline tangent of an express highway follows the line 4 x − 3 y + 15 = 0 4x - 3y + 15 = 0 4 x − 3 y + 15 = 0 . A circular stormwater detention basin has a perimeter defined by the equation: x 2 + y 2 − 6 x − 8 y − 11 = 0 x^2 + y^2 - 6x - 8y - 11 = 0 x 2 + y 2 − 6 x − 8 y − 11 = 0 Determine:
The coordinates of the center ( h , k ) (h, k) ( h , k ) and the radius R R R of the retention basin.
The shortest clearance distance from the center of the retention basin to the centerline tangent of the highway.
Does the highway tangent intersect the retention basin, touch it as a tangent, or clear it completely?
Solution:
Reduce the circle equation to standard form by completing the square: ( x 2 − 6 x + 9 ) + ( y 2 − 8 y + 16 ) = 11 + 9 + 16 (x^2 - 6x + 9) + (y^2 - 8y + 16) = 11 + 9 + 16 ( x 2 − 6 x + 9 ) + ( y 2 − 8 y + 16 ) = 11 + 9 + 16 ( x − 3 ) 2 + ( y − 4 ) 2 = 36 (x - 3)^2 + (y - 4)^2 = 36 ( x − 3 ) 2 + ( y − 4 ) 2 = 36
- Center of the basin: ( h , k ) = ( 3 , 4 ) (h, k) = (3, 4) ( h , k ) = ( 3 , 4 )
- Radius of the basin: R = 36 = 6.0 m R = \sqrt{36} = 6.0\text{ m} R = 36 = 6.0 m
Compute the perpendicular distance from center ( 3 , 4 ) (3, 4) ( 3 , 4 ) to the line 4 x − 3 y + 15 = 0 4x - 3y + 15 = 0 4 x − 3 y + 15 = 0 : d = ∣ A x 1 + B y 1 + C ∣ A 2 + B 2 = ∣ 4 ( 3 ) − 3 ( 4 ) + 15 ∣ 4 2 + ( − 3 ) 2 = ∣ 12 − 12 + 15 ∣ 16 + 9 = 15 5 = 3.0 m d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} = \frac{|4(3) - 3(4) + 15|}{\sqrt{4^2 + (-3)^2}} = \frac{|12 - 12 + 15|}{\sqrt{16 + 9}} = \frac{15}{5} = 3.0\text{ m} d = A 2 + B 2 ∣ A x 1 + B y 1 + C ∣ = 4 2 + ( − 3 ) 2 ∣4 ( 3 ) − 3 ( 4 ) + 15∣ = 16 + 9 ∣12 − 12 + 15∣ = 5 15 = 3.0 m
Evaluate clearance relationship: Basin radius R = 6.0 m > Distance to centerline d = 3.0 m \text{Basin radius } R = 6.0\text{ m} > \text{Distance to centerline } d = 3.0\text{ m} Basin radius R = 6.0 m > Distance to centerline d = 3.0 m Because the perpendicular distance from the center to the line ( 3.0 m 3.0\text{ m} 3.0 m ) is strictly less than the radius of the circle ( 6.0 m 6.0\text{ m} 6.0 m ), the highway centerline intersects the retention basin in a secant line at two distinct points , cutting through the stormwater facility.
CELE Board Exam Traps & Strategic Checklists
Warning
Focal Parameter Relations: Do not confuse the focal relationship for ellipses and hyperbolas:
For an ellipse : a 2 = b 2 + c 2 ⟹ c = a 2 − b 2 a^2 = b^2 + c^2 \implies c = \sqrt{a^2 - b^2} a 2 = b 2 + c 2 ⟹ c = a 2 − b 2 (major axis a a a is the longest semi-axis).
For a hyperbola : c 2 = a 2 + b 2 ⟹ c = a 2 + b 2 c^2 = a^2 + b^2 \implies c = \sqrt{a^2 + b^2} c 2 = a 2 + b 2 ⟹ c = a 2 + b 2 (focal distance c c c is the longest parameter).
Asymptote Slopes: In a hyperbola with a horizontal transverse axis, the asymptotes are y − k = ± b a ( x − h ) y - k = \pm \frac{b}{a}(x - h) y − k = ± a b ( x − h ) . But if the transverse axis is vertical, the asymptotes flip to y − k = ± a b ( x − h ) y - k = \pm \frac{a}{b}(x - h) y − k = ± b a ( x − h ) . Check the orientation of the positive squared term first!
Latus Rectum Formula: The length of the latus rectum for both ellipses and hyperbolas is L R = 2 b 2 a LR = \frac{2b^2}{a} L R = a 2 b 2 . For a parabola, it is simply ∣ 4 p ∣ |4p| ∣4 p ∣ .