2.3 Analytic & Solid Geometry

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.

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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): $$x = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}, \quad y = \frac{r_2 y_1 + r_1 y_2}{r_1 + r_2}$$

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: $$\tan \phi = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$$

Standard Forms of Line Equations

1. Slope-Intercept Form: $y = mx + b$
2. Point-Slope Form: $y - y_1 = m(x - x_1)$
3. Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$
4. General Linear Form: $Ax + By + C = 0$

Perpendicular Distance from Point $(x_1, y_1)$ to Line $Ax + By + C = 0$

$$d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^2 + B^2}}$$

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2. Conic Sections

The general second-degree equation in two variables is:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

Classification via Discriminant ($B^2 - 4AC$)

Assuming no degenerate cases:

Standard Equations & Geometric Properties

1. Circle

$$(x - h)^2 + (y - k)^2 = r^2$$

• 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

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a > b)$$

• 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

$$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$

• Foci distance: $c = \sqrt{a^2 + b^2}$
• Eccentricity: $e = \frac{c}{a} > 1$
• Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$

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3. Polar Coordinate Transformations

Polar coordinates represent a point as $(r, \theta)$, where $r$ is the radial distance and $\theta$ is the polar angle.

$$\text{Cartesian to Polar: } r = \sqrt{x^2 + y^2}, \quad \theta = \arctan\left(\frac{y}{x}\right)$$ $$\text{Polar to Cartesian: } x = r \cos \theta, \quad y = r \sin \theta$$

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4. Solid Geometry & 3D Mensuration

Common 3D Solids Formulas

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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)

$$\text{Surface Area } S = 2\pi \bar{y} L$$

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)

$$\text{Volume } V = 2\pi \bar{y} A$$

where:

• $A$ = area of the generating plane region.
• $\bar{y}$ = distance from the axis of rotation to the centroid of the plane region.

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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:

1. Identify coefficients: $A = 4$, $B = -3$, $C = 7$, $x_1 = 3$, $y_1 = -2$.
2. Apply the distance formula: $$d = \frac{|A x_1 + B y_1 + C|}{\sqrt{A^2 + B^2}} = \frac{|4(3) + (-3)(-2) + 7|}{\sqrt{4^2 + (-3)^2}}$$
3. Evaluate numerator and denominator: $$\text{Numerator} = |12 + 6 + 7| = 25$$ $$\text{Denominator} = \sqrt{16 + 9} = \sqrt{25} = 5$$ $$d = \frac{25}{5} = 5\ \text{units}$$

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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:

1. Identify plane region area $A$ and centroid distance $\bar{y}$: $$A = \pi r^2 = \pi (3)^2 = 9\pi\ \text{cm}^2$$ $$\bar{y} = R = 10\ \text{cm}$$
2. Apply Pappus's Second Theorem: $$V = 2\pi \bar{y} A = 2\pi (10) (9\pi) = 180\pi^2 \approx 1776.5288\ \text{cm}^3$$