2.3 Analytic & Solid Geometry
Key Takeaways
- Straight-line geometry establishes slope relationships (m_1 = m_2 for parallel, m_1 m_2 = -1 for perpendicular) and point-to-line perpendicular distance d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}.
- 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, \theta) convert to Cartesian via x = r\cos\theta and y = r\sin\theta, 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\pi \bar{y} L) and volumes (V = 2\pi \bar{y} A) of revolution by rotating 2D curves or plane regions about a coplanar non-intersecting axis.
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 $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$:
- Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
- Midpoint Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
- Point of Division ($r_1 : r_2$ ratio):
Line Slope & Angle Between Lines
The slope $m$ of a non-vertical line is $m = \frac{y_2 - y_1}{x_2 - x_1} = \tan \theta$.
- Parallel Lines: $m_1 = m_2$
- Perpendicular Lines: $m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$
- Acute Angle $\phi$ Between Lines:
Standard Forms of Line Equations
- Slope-Intercept Form: $y = mx + b$
- Point-Slope Form: $y - y_1 = m(x - x_1)$
- Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$
- General Linear Form: $Ax + By + C = 0$
Perpendicular Distance from Point $(x_1, y_1)$ to Line $Ax + By + C = 0$
2. Conic Sections
The general second-degree equation in two variables is:
Classification via Discriminant ($B^2 - 4AC$)
Assuming no degenerate cases:
| Discriminant $B^2 - 4AC$ | Conic Type | Eccentricity ($e$) |
|---|---|---|
| $B^2 - 4AC < 0$ | Ellipse (Circle if $A = C, B = 0$) | $0 \le e < 1$ ($e = 0$ for Circle) |
| $B^2 - 4AC = 0$ | Parabola | $e = 1$ |
| $B^2 - 4AC > 0$ | Hyperbola | $e > 1$ |
Standard Equations & Geometric Properties
1. Circle
- Center: $(h, k)$, Radius: $r = \sqrt{h^2 + k^2 - F}$ for general form $x^2 + y^2 + Dx + Ey + F = 0$ where $h = -D/2$, $k = -E/2$.
2. Parabola
- Vertical Axis: $(x - h)^2 = 4p(y - k)$ (Opens UP if $p>0$, DOWN if $p<0$)
- Horizontal Axis: $(y - k)^2 = 4p(x - h)$ (Opens RIGHT if $p>0$, LEFT if $p<0$)
- Focal distance: $|p|$, Latus Rectum length: $L = |4p|$.
3. Ellipse
- Foci distance: $c = \sqrt{a^2 - b^2}$
- Eccentricity: $e = \frac{c}{a} < 1$
- Latus Rectum length: $L = \frac{2b^2}{a}$
- Area of Ellipse: $\text{Area} = \pi a b$
4. Hyperbola
- Foci distance: $c = \sqrt{a^2 + b^2}$
- Eccentricity: $e = \frac{c}{a} > 1$
- Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$
3. Polar Coordinate Transformations
Polar coordinates represent a point as $(r, \theta)$, where $r$ is the radial distance and $\theta$ is the polar angle.
4. Solid Geometry & 3D Mensuration
Common 3D Solids Formulas
| Solid | Volume ($V$) | Lateral Surface Area ($A_L$) | Total Surface Area ($A_T$) |
|---|---|---|---|
| Right Circular Cylinder | $V = \pi r^2 h$ | $A_L = 2\pi r h$ | $A_T = 2\pi r h + 2\pi r^2$ |
| Right Circular Cone | $V = \frac{1}{3}\pi r^2 h$ | $A_L = \pi r L$ ($L = \sqrt{r^2 + h^2}$) | $A_T = \pi r L + \pi r^2$ |
| Sphere | $V = \frac{4}{3}\pi r^3$ | — | $A_T = 4\pi r^2$ |
| Frustum of Cone | $V = \frac{1}{3}\pi h (r_1^2 + r_2^2 + r_1 r_2)$ | $A_L = \pi (r_1 + r_2) L$ | $A_T = A_L + \pi r_1^2 + \pi r_2^2$ |
| Pyramid | $V = \frac{1}{3} A_{\text{base}} h$ | $\frac{1}{2} P_{\text{base}} L$ | $A_L + A_{\text{base}}$ |
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)
where:
- $L$ = length of the generating curve arc.
- $\bar{y}$ = distance from the axis of rotation to the centroid of the curve.
Second Theorem (Volume of Revolution)
where:
- $A$ = area of the generating plane region.
- $\bar{y}$ = 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 $P(3, -2)$ to the line $4x - 3y + 7 = 0$.
Solution:
- Identify coefficients: $A = 4$, $B = -3$, $C = 7$, $x_1 = 3$, $y_1 = -2$.
- Apply the distance formula:
- Evaluate numerator and denominator:
Example 2: Volume of Solid Torus via Pappus's Second Theorem
Problem: A circle of radius $r = 3\ \text{cm}$ is rotated about an axis located $R = 10\ \text{cm}$ from the center of the circle. Calculate the volume of the resulting solid torus.
Solution:
- Identify plane region area $A$ and centroid distance $\bar{y}$:
- Apply Pappus's Second Theorem:
What is the perpendicular distance from the point $(3, -2)$ to the line $4x - 3y + 7 = 0$?
An ellipse has the equation $9x^2 + 25y^2 = 225$. What is its eccentricity $e$?
A circle of radius $r = 3\ \text{cm}$ is rotated about an axis in its plane located $R = 10\ \text{cm}$ from the center of the circle. What is the volume of the solid torus generated?