Core Concepts in Analytical Geometry: Lines, Conics, and Algebra

Slope, Inclination, and Lines

PENDING

Slope

Inclination and Tilt

This refers to the straight tilt or angle of a line.

Parallel Lines Definition

Two lines are parallel if their slopes are equal: L₁ || L₂ if m₁ = m₂.

To find the angle (theta), the slope is used, often in conjunction with trigonometric tables or functions.

Perpendicular Lines Definition

Two lines are perpendicular if they form an angle of 90⁰. This occurs when the product of their slopes is -1:

(m₁)(m₂) = -1

General Equation of a Line

To formulate the general equation of a line, two components are typically needed:

• The slope (m)
• A specific point on the line (x₁, y₁)

The variables (x, y) in the equation represent any point on that line.

Midpoint of a Line Segment

The coordinates of the midpoint of a line segment are represented as (x, y).

Equation of an Intersecting Line

The equation of a line (which can intersect other lines) is typically obtained if the slope (pendiente) is known and specific points on the line are identified or used.

A point on the line can be denoted as P(x, y).

Distance from a Point to a Line

A specific formula is used to calculate the distance from a point to a line.

This often involves the slope-line equation of the line (where m = slope).

To apply the formula, you might need to rearrange the line's equation, for instance, by solving for Y. When dealing with fractions, it's important to use a common denominator. For example:

Circle with Center (h, k)

Cartesian Equation of the Circle:

The points (h, k) represent the coordinates of the center of the circle. In the standard Cartesian equation of a circle, (x-h)² + (y-k)² = r², the values h and k appear with signs opposite to their actual coordinate values.

Rule for Expanding Binomials (Example):

(x-5)² = x² - 2(x)(5) + 5² = x² - 10x + 25

Note: In the expansion of (x-5)², the middle term (-10x) results from 2 * x * (-5). The constant term (+25) is (-5)². When completing the square to find the center/radius from a general form, the coefficient of the x term (e.g., -10) is halved (-5) and then squared (+25) to find the constant part of the binomial square.

Very important: To obtain the third term (the constant) when expanding a binomial square like (a ± b)², you square the second term of the binomial (b²).

This expansion is a development of the general formula.

Parabola: Vertex at Origin

There are four standard cases for a parabola with its vertex at the origin (0,0):

• Opens Up: Focus (0, p); Directrix Equation: y = -p
• Opens Right: Focus (p, 0); Directrix Equation: x = -p
• Opens Left: Focus (-p, 0); Directrix Equation: x = p
• Opens Down: Focus (0, -p); Directrix Equation: y = p

Parabola: Vertex at (h, k)

When the vertex of the parabola is at any point (h, k), and 'P' is the focal distance (distance from vertex to focus and vertex to directrix):

Parabola Opens Up

• Focus: (h, k + P)
• Latus Rectum (LR) length: |4P|
• Directrix: y = k - P

Parabola Opens Downward

• Focus: (h, k - P)
• Latus Rectum (LR) length: |4P|
• Directrix: y = k + P

Parabola Opens Right

• Focus: (h + P, k)
• Latus Rectum (LR) length: |4P|
• Directrix: x = h - P

Parabola Opens to the Left

• Focus: (h - P, k)
• Latus Rectum (LR) length: |4P|
• Directrix as stated in original text: x = h - P
  (Correction/Clarification: For a parabola opening left with focus (h - P, k), the standard directrix is x = h + P, assuming P > 0 is the focal distance.)

Ellipse Properties and Equations

Ellipse: Center at Origin, Foci on X-Axis

The equation for an ellipse centered at the origin (0,0) with its foci on the x-axis.

• Major axis length = 2a
• Minor axis length = 2b
• Foci: (c, 0) and (-c, 0)
• Vertices as listed in original text: (0, b), (0, -a), (0, -b), (0, a)
  (Correction/Clarification: This list appears to describe points solely on the y-axis. For an ellipse with foci on the x-axis, the standard major vertices are (±a, 0) and co-vertices on the minor axis are (0, ±b). 'a' is the semi-major axis length, 'b' is the semi-minor axis length, and a > b.)
• Eccentricity (e): (Formula: e = c/a)
• Parameter c: (Formula: c² = a² - b²)
• Latus Rectum (LR) length: (Formula: LR = 2b²/a)

Ellipse: Center at Origin, Foci on Y-Axis

The equation for an ellipse centered at the origin (0,0) with its foci on the y-axis.

• Major axis length = 2a
• Minor axis length = 2b
• Foci: (0, c) and (0, -c)
• Vertices as listed in original text: (0, b), (0, a), (0, -b), (0, a)
  (Correction/Clarification: This list contains points on the y-axis. For an ellipse with foci on the y-axis, the standard major vertices are (0, ±a) and co-vertices on the minor axis are (±b, 0). 'a' is the semi-major axis length, 'b' is the semi-minor axis length, and a > b. The point (0,a) is repeated in the original list.)

Basic Fraction Operations

Key operations with fractions include:

• Addition:
• Subtraction:
• Multiplication:

Rule of Signs in Multiplication

The rules for signs when multiplying numbers are:

• (-) × (-) = +
• (+) × (+) = +
• (-) × (+) = -
• (+) × (-) = -

Equation Balancing: Inverse Operations

When moving a term or operation to the other side of an equals sign in an equation (to maintain balance):

• Subtraction (-) on one side becomes Addition (+) on the other.
• Addition (+) on one side becomes Subtraction (-) on the other.
• Division (/) on one side becomes Multiplication (*) on the other.
• Multiplication (*) on one side becomes Division (/) on the other.