Study guide
2.3 Analytic & Solid Geometry
PRC Registered Electrical Engineer REE
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
2.3 Analytic & Solid Geometry
Key Takeaways Explain This Topic AI
Straight-line geometry establishes slope relationships ( m 1 = m 2 m_1 = m_2 m 1 = m 2 for parallel, m 1 m 2 = − 1 m_1 m_2 = -1 m 1 m 2 = − 1 for perpendicular) and point-to-line perpendicular distance d = ∣ A x 1 + B y 1 + C ∣ A 2 + B 2 d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ A x 1 + B y 1 + C ∣ .
Conic sections are classified by discriminant B^2 - 4AC (parabola = 0, ellipse < 0, hyperbola > 0) and eccentricity e = c/a (e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1 hyperbola).
Polar coordinates ( r , θ ) (r, \theta) ( r , θ ) convert to Cartesian via x = r cos θ x = r\cos\theta x = r cos θ and y = r sin θ y = r\sin\theta y = r sin θ , facilitating the representation of circular and radial physical systems.
Mensuration formulas for 3D polyhedrons, cylinders, cones, and spheres determine volumes, lateral surface areas, and frustum dimensions.
Pappus's Centroid Theorems calculate surface areas ( S = 2 π y ˉ L S = 2\pi \bar{y} L S = 2 π y ˉ L ) and volumes ( V = 2 π y ˉ A V = 2\pi \bar{y} A V = 2 π y ˉ A ) of revolution by rotating 2D curves or plane regions about a coplanar non-intersecting axis.
Last updated: August 2026
2.3 Analytic & Solid Geometry
Analytic geometry provides coordinate-based tools for analyzing lines, curves, and spatial surfaces, while solid geometry (mensuration) deals with three-dimensional solids. For the PRC Registered Electrical Engineer (REE) examination, mastery of 2D line and conic equations, polar coordinate transformations, 3D surface areas/volumes, and Pappus's Centroid Theorems is essential.
- 1. Analytic Geometry of Straight Lines
- Basic Coordinate Formulas
For two points P 1 ( x 1 , y 1 ) P_1(x_1, y_1) P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ) P_2(x_2, y_2) P 2 ( x 2 , y 2 ) :
Distance Formula: 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
Midpoint Formula: M = ( x 1 + x 2 2 , y 1 + y 2 2 ) M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) M = ( 2 x 1 + x 2 , 2 y 1 + y 2 )
Point of Division ( r 1 : r 2 r_1 : r_2 r 1 : r 2 ratio): x = r 2 x 1 + r 1 x 2 r 1 + r 2 , y = r 2 y 1 + r 1 y 2 r 1 + r 2 x = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}, \quad y = \frac{r_2 y_1 + r_1 y_2}{r_1 + r_2} x = r 1 + r 2 r 2 x 1 + r 1 x 2 , y = r 1 + r 2 r 2 y 1 + r 1 y 2
Line Slope & Angle Between Lines
The slope m m m of a non-vertical line is 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 .
Parallel Lines: m 1 = m 2 m_1 = m_2 m 1 = m 2
Perpendicular Lines: m 1 ⋅ m 2 = − 1 ⟹ m 2 = − 1 m 1 m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1} m 1 ⋅ m 2 = − 1 ⟹ m 2 = − m 1 1
Acute Angle ϕ \phi ϕ Between Lines: tan ϕ = ∣ m 2 − m 1 1 + m 1 m 2 ∣ \tan \phi = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| tan ϕ = 1 + m 1 m 2 m 2 − m 1
- Standard Forms of Line Equations
- Slope-Intercept Form: y = m x + b y = mx + b y = m x + b
- Point-Slope Form: y − y 1 = m ( x − x 1 ) y - y_1 = m(x - x_1) y − y 1 = m ( x − x 1 )
- Intercept Form: x a + y b = 1 \frac{x}{a} + \frac{y}{b} = 1 a x + b y = 1
- General Linear Form: A x + B y + C = 0 Ax + By + C = 0 A x + B y + C = 0
Perpendicular Distance from Point ( x 1 , y 1 ) (x_1, y_1) ( x 1 , y 1 ) to Line A x + B y + C = 0 Ax + By + C = 0 A x + B y + C = 0 d = ∣ A x 1 + B y 1 + C ∣ A 2 + B 2 d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^2 + B^2}} d = A 2 + B 2 ∣ A x 1 + B y 1 + C ∣
2. Conic Sections
The general second-degree equation in two variables is: 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
Classification via Discriminant ( B 2 − 4 A C B^2 - 4AC B 2 − 4 A C )
Assuming no degenerate cases:
Discriminant B 2 − 4 A C B^2 - 4AC B 2 − 4 A C Conic Type Eccentricity ( e e e )
B 2 − 4 A C < 0 B^2 - 4AC < 0 B 2 − 4 A C < 0 Ellipse (Circle if A = C , B = 0 A = C, B = 0 A = C , B = 0 ) 0 ≤ e < 1 0 \le e < 1 0 ≤ e < 1 ( e = 0 e = 0 e = 0 for Circle)
- B 2 − 4 A C = 0 B^2 - 4AC = 0 B 2 − 4 A C = 0 Parabola e = 1 e = 1 e = 1
- B 2 − 4 A C > 0 B^2 - 4AC > 0 B 2 − 4 A C > 0 Hyperbola e > 1 e > 1 e > 1
- Standard Equations & Geometric Properties
- 1. Circle ( x − h ) 2 + ( y − k ) 2 = r 2 (x - h)^2 + (y - k)^2 = r^2 ( x − h ) 2 + ( y − k ) 2 = r 2
Center: ( h , k ) (h, k) ( h , k ) , Radius: r = h 2 + k 2 − F r = \sqrt{h^2 + k^2 - F} r = h 2 + k 2 − F for 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 .
2. Parabola
Vertical Axis: ( x − h ) 2 = 4 p ( y − k ) (x - h)^2 = 4p(y - k) ( x − h ) 2 = 4 p ( y − k ) (Opens UP if p > 0 p>0 p > 0 , DOWN if p < 0 p<0 p < 0 )
Horizontal Axis: ( y − k ) 2 = 4 p ( x − h ) (y - k)^2 = 4p(x - h) ( y − k ) 2 = 4 p ( x − h ) (Opens RIGHT if p > 0 p>0 p > 0 , LEFT if p < 0 p<0 p < 0 )
Focal distance: ∣ p ∣ |p| ∣ p ∣ , Latus Rectum length: L = ∣ 4 p ∣ L = |4p| L = ∣4 p ∣ .
3. Ellipse ( 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 distance: c = a 2 − b 2 c = \sqrt{a^2 - b^2} c = a 2 − b 2
- Eccentricity: e = c a < 1 e = \frac{c}{a} < 1 e = a c < 1
- Latus Rectum length: L = 2 b 2 a L = \frac{2b^2}{a} L = a 2 b 2
- Area of Ellipse: Area = π a b \text{Area} = \pi a b Area = πab
4. Hyperbola ( 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 distance: c = a 2 + b 2 c = \sqrt{a^2 + b^2} c = a 2 + b 2
- Eccentricity: e = c a > 1 e = \frac{c}{a} > 1 e = a c > 1
- Asymptotes: y − k = ± b a ( x − h ) y - k = \pm \frac{b}{a}(x - h) y − k = ± a b ( x − h )
- 3. Polar Coordinate Transformations
Polar coordinates represent a point as ( r , θ ) (r, \theta) ( r , θ ) , where r r r is the radial distance and θ \theta θ is the polar angle. Cartesian to Polar: r = x 2 + y 2 , θ = arctan ( y x ) \text{Cartesian to Polar: } r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\left(\frac{y}{x}\right) Cartesian to Polar: r = x 2 + y 2 , θ = arctan ( x y ) Polar to Cartesian: x = r cos θ , y = r sin θ \text{Polar to Cartesian: } x = r \cos \theta, \quad y = r \sin \theta Polar to Cartesian: x = r cos θ , y = r sin θ
- 4. Solid Geometry & 3D Mensuration
- Common 3D Solids Formulas
Solid Volume ( V V V ) Lateral Surface Area ( A L A_L A L ) Total Surface Area ( A T A_T A T )
Right Circular Cylinder V = π r 2 h V = \pi r^2 h V = π r 2 h A L = 2 π r h A_L = 2\pi r h A L = 2 π r h A T = 2 π r h + 2 π r 2 A_T = 2\pi r h + 2\pi r^2 A T = 2 π r h + 2 π r 2
Right Circular Cone V = 1 3 π r 2 h V = \frac{1}{3}\pi r^2 h V = 3 1 π r 2 h A L = π r L A_L = \pi r L A L = π r L ( L = r 2 + h 2 L = \sqrt{r^2 + h^2} L = r 2 + h 2 ) A T = π r L + π r 2 A_T = \pi r L + \pi r^2 A T = π r L + π r 2
Sphere V = 4 3 π r 3 V = \frac{4}{3}\pi r^3 V = 3 4 π r 3 — A T = 4 π r 2 A_T = 4\pi r^2 A T = 4 π r 2
Frustum of Cone V = 1 3 π h ( r 1 2 + r 2 2 + r 1 r 2 ) V = \frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2) V = 3 1 πh ( r 1 2 + r 2 2 + r 1 r 2 ) A L = π ( r 1 + r 2 ) L A_L = \pi (r_1 + r_2) L A L = π ( r 1 + r 2 ) L A T = A L + π r 1 2 + π r 2 2 A_T = A_L + \pi r_1^2 + \pi r_2^2 A T = A L + π r 1 2 + π r 2 2
Pyramid V = 1 3 A base h V = \frac{1}{3} A_{\text{base}} h V = 3 1 A base h 1 2 P base L \frac{1}{2} P_{\text{base}} L 2 1 P base L A L + A base A_L + A_{\text{base}} A L + A base
5. Pappus's Centroid Theorems
Pappus's Theorems provide an elegant way to calculate surface areas and volumes of solids formed by rotating plane figures about a non-intersecting coplanar axis.
First Theorem (Surface Area of Revolution) Surface Area S = 2 π y ˉ L \text{Surface Area } S = 2\pi \bar{y} L Surface Area S = 2 π y ˉ L
where:
L L L = length of the generating curve arc.
y ˉ \bar{y} y ˉ = distance from the axis of rotation to the centroid of the curve.
Second Theorem (Volume of Revolution) Volume V = 2 π y ˉ A \text{Volume } V = 2\pi \bar{y} A Volume V = 2 π y ˉ A
where:
A A A = area of the generating plane region.
y ˉ \bar{y} y ˉ = distance from the axis of rotation to the centroid of the plane region.
- Solved Practice Examples
- Example 1: Point-to-Line Distance
Problem: Compute the perpendicular distance from point P ( 3 , − 2 ) P(3, -2) P ( 3 , − 2 ) to the line 4 x − 3 y + 7 = 0 4x - 3y + 7 = 0 4 x − 3 y + 7 = 0 .
Solution:
Identify coefficients: A = 4 A = 4 A = 4 , B = − 3 B = -3 B = − 3 , C = 7 C = 7 C = 7 , x 1 = 3 x_1 = 3 x 1 = 3 , y 1 = − 2 y_1 = -2 y 1 = − 2 .
Apply the distance formula: d = ∣ A x 1 + B y 1 + C ∣ A 2 + B 2 = ∣ 4 ( 3 ) + ( − 3 ) ( − 2 ) + 7 ∣ 4 2 + ( − 3 ) 2 d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^2 + B^2}} = \frac{|4(3) + (-3)(-2) + 7|}{\sqrt{4^2 + (-3)^2}} d = A 2 + B 2 ∣ A x 1 + B y 1 + C ∣ = 4 2 + ( − 3 ) 2 ∣4 ( 3 ) + ( − 3 ) ( − 2 ) + 7∣
Evaluate numerator and denominator: Numerator = ∣ 12 + 6 + 7 ∣ = 25 \text{Numerator} = |12 + 6 + 7| = 25 Numerator = ∣12 + 6 + 7∣ = 25 Denominator = 16 + 9 = 25 = 5 \text{Denominator} = \sqrt{16 + 9} = \sqrt{25} = 5 Denominator = 16 + 9 = 25 = 5 d = 25 5 = 5 units d = \frac{25}{5} = 5\ \text{units} d = 5 25 = 5 units
Example 2: Volume of Solid Torus via Pappus's Second Theorem
Problem: A circle of radius r = 3 cm r = 3\ \text{cm} r = 3 cm is rotated about an axis located R = 10 cm R = 10\ \text{cm} R = 10 cm from the center of the circle. Calculate the volume of the resulting solid torus.
Solution:
Identify plane region area A A A and centroid distance y ˉ \bar{y} y ˉ : A = π r 2 = π ( 3 ) 2 = 9 π cm 2 A = \pi r^2 = \pi (3)^2 = 9\pi\ \text{cm}^2 A = π r 2 = π ( 3 ) 2 = 9 π cm 2 y ˉ = R = 10 cm \bar{y} = R = 10\ \text{cm} y ˉ = R = 10 cm
Apply Pappus's Second Theorem: V = 2 π y ˉ A = 2 π ( 10 ) ( 9 π ) = 180 π 2 ≈ 1776.5288 cm 3 V = 2\pi \bar{y} A = 2\pi (10) (9\pi) = 180\pi^2 \approx 1776.5288\ \text{cm}^3 V = 2 π y ˉ A = 2 π ( 10 ) ( 9 π ) = 180 π 2 ≈ 1776.5288 cm 3