Coordinate Geometry

A straight line is the shortest distance between two points. In coordinate geometry, it is described as a relationship between x and y.

Distance and Section Formula

The Distance Formula: Used to find the length between two points P(x₁, y₁) and Q(x₂, y₂).

• Example 1: Distance between (0,0) and (3,4) is √(3² + 4²) = 5 units.
• Example 2: Distance between (-1, -1) and (2, 3) is √((2+1)² + (3+1)²) = √(9+16) = 5 units.

The Section Formula: Used to find a point that divides a line segment in a specific ratio (m:n).

• Example 1: Midpoint of (2,4) and (6,8) [Ratio 1:1] = ((2+6)/2, (4+8)/2) = (4, 6).
• Example 2: Point dividing (1,2) and (4,8) in ratio 2:1. x = (2(4)+1(1))/3 = 3, y = (2(8)+1(2))/3 = 6. Point: (3,6).

Slope (Gradient)

The slope (m) represents the steepness of a line (tan θ).

• Horizontal lines: Slope = 0.
• Vertical lines: Undefined slope.
• Parallel lines: m₁ = m₂.
• Perpendicular lines: m₁ × m₂ = -1.
• Example: Line through (1,2) and (3,6) has m = (6-2)/(3-1) = 4/2 = 2.

Equations of a Straight Line

• Slope-Intercept Form: y = mx + c
• Point-Slope Form: y - y₁ = m(x - x₁)
• Two-Point Form: y - y₁ = ((y₂-y₁)/(x₂-x₁))(x - x₁)
• General Equation: Ax + By + C = 0

Real-Life Use: Civil Engineering & Road Planning. Engineers use these equations to calculate the grade (slope) of highways to ensure safe climbing and effective water drainage.

Conic sections are curves obtained by slicing a double-cone with a plane.

The Circle

Points equidistant from a fixed center (h, k).

• Example: Center (2, -3), radius 4 → (x-2)² + (y+3)² = 16.

The Parabola

Points equidistant from a fixed point (Focus) and a fixed line (Directrix).

• Example: y² = 12x → 4a = 12 → a = 3. Focus is (3,0).

The Ellipse

Points where the sum of distances from two fixed foci is constant.

• Example: x²/25 + y²/9 = 1. Major axis = 10, Minor axis = 6.

The Hyperbola

Points where the difference of distances from two foci is constant.

• Example: x²/16 - y²/9 = 1.

Real-Life Use: Parabolas are used in satellite dishes; Ellipses describe planetary orbits; Hyperbolas are used in LORAN navigation systems.

3D Geometry adds the Z-axis (height/depth), creating three planes: XY, YZ, and ZX.

Coordinates and Distance

A point is represented as (x, y, z).

• Example: Distance between (0,0,0) and (1,2,2) = √(1² + 2² + 2²) = √9 = 3.

3D Section Formula

• Example: Midpoint of (0,0,0) and (2,4,6) is (1, 2, 3).

Real-Life Use: 3D Printing and CGI. Printer heads move in X, Y, and Z coordinates to create physical objects based on 3D mathematical equations.

Comparison of Conic Sections

Modern Tools & Industry Applications

• AutoCAD: Used by architects for precise geometry.
• MATLAB/Python: Used for visualizing 3D surfaces.
• GPS: Uses "Trilateration" (intersecting 3D spheres) to locate users.

Scenario: An engineering firm is designing a bridge with three geometric components:

1. The Straight Line Component: The bridge deck starts at A(10, 20) and ends at B(110, 20). Using the Distance Formula, length = √((110-10)² + (20-20)²) = 100m. A slope of 0 ensures the deck is level.

2. The Conic Section Component: Support cables follow a Parabolic path (x² = 4ay). The vertex is the lowest point. For entrance arches, an Ellipse is used for symmetry.

3. The 3D Geometry Component: A monitoring drone hovers at (50, 20, 30). Distance to start point A(10, 20, 0) is calculated via 3D Distance Formula: √((50-10)² + (20-20)² + (30-0)²) = √ (1600 + 900) = 50m.

1. The slope of a line perpendicular to y = 3x + 5 is:
   a) 3 b) -3 c) 1/3 d) -1/3
2. The distance between points (3,0) and (0,4) is:
   a) 7 b) 5 c) 1 d) 25
3. The eccentricity of a circle is:
   a) 1 b) 0 c) ∞ d) 0.5
4. The standard equation y² = 4ax represents a:
   a) Circle b) Ellipse c) Parabola d) Hyperbola
5. The midpoint of (2, -3) and (4, 7) is:
   a) (3, 2) b) (6, 4) c) (1, 5) d) (3, 5)
6. Which conic section has e > 1?
   a) Circle b) Parabola c) Ellipse d) Hyperbola
7. The distance of point (1, 2, 2) from the origin (0, 0, 0) is:
   a) 3 b) 9 c) √5 d) 5
8. If a line is parallel to the x-axis, its slope is:
   a) Undefined b) 1 c) 0 d) -1
9. The sum of distances from two foci is constant in an:
   a) Hyperbola b) Ellipse c) Parabola d) Circle
10. The point (0, 0, 0) in 3D geometry is called the:
    a) Vertex b) Focus c) Origin d) Center

Answers: 1(d), 2(b), 3(b), 4(c), 5(a), 6(d), 7(a), 8(c), 9(b), 10(c).

1. Q: Derive the distance formula between two points in 2D.
   A: Using Pythagoras Theorem, for points (x₁, y₁) and (x₂, y₂), the horizontal distance is (x₂-x₁) and vertical is (y₂-y₁). Hypotenuse² = Base² + Perpendicular². Thus, d = √((x₂-x₁)² + (y₂-y₁)²).
2. Q: Find the equation of a line that passes through (1, 2) and is parallel to y = 4x - 9.
   A: Parallel lines have the same slope, so m = 4. Using point-slope form: y - 2 = 4(x - 1) → y = 4x - 2.
3. Q: What is the difference between a Circle and an Ellipse?
   A: A circle has one center and constant radius (e=0). An ellipse has two foci and the sum of distances to these foci is constant (0 < e < 1).
4. Q: Find the coordinates of the point which divides the line joining (1, 2, 3) and (4, 5, 6) in the ratio 1:1.
   A: Midpoint = ((1+4)/2, (2+5)/2, (3+6)/2) = (2.5, 3.5, 4.5).
5. Q: Find the focus and directrix of y² = 16x.
   A: 4a = 16 → a = 4. Focus = (a, 0) = (4, 0). Directrix: x = -a → x = -4.
6. Q: If the slope of a line is tan(45°), what is the equation of the line passing through the origin?
   A: tan(45°) = 1. Since it passes through (0,0), y = mx → y = 1x or y = x.
7. Q: Explain the concept of Eccentricity.
   A: Eccentricity (e) measures how much a conic section deviates from a circle: e=0 (Circle), e=1 (Parabola), e < 1 (Ellipse), e > 1 (Hyperbola).
8. Q: Find the distance between P(1, -1, 3) and Q(2, 3, -5).
   A: d = √((2-1)² + (3+1)² + (-5-3)²) = √(1² + 4² + (-8)²) = √81 = 9.
9. Q: Find the equation of a circle with center (0,0) and passing through (3,4).
   A: Radius r = √(3² + 4²) = 5. Equation: x² + y² = 5² → x² + y² = 25.
10. Q: A line L₁ has slope m₁. If L₂ is perpendicular to L₁, what is the relation between their slopes?
    A: The product of their slopes is -1 (m₁ × m₂ = -1).

• Slope Direction: Slope = Steepness. Formula is ΔY / ΔX.
• Conic Eccentricity (e):
  • Circle: Cero (Zero) → e=0
  • Parabola: Precisely One → e=1
  • Ellipse: Even less than one → e < 1
  • Hyperbola: Huge (more than one) → e > 1
• Distance Formula Trick: "Subtract, Square, Sum, Square-root" (S4).