Coordinate Geometry

• Introduction
  • Definition: Geometry using a coordinate system
  • Real‑life uses: navigation, engineering, computer graphics
• Coordinate System
  • Cartesian Plane
    • Axes: x‑axis (horizontal), y‑axis (vertical)
    • Origin O(0, 0)
  • Quadrants
    • I (+,+) II (−,+) III (−,−) IV (+,−)
• Points & Distance
  • Distance Formula
    • Derivation from Pythagoras:
      $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
    • Example: Distance between $A(2,3)$ and $B(7,11)$
      $d = \sqrt{(7-2)^2+(11-3)^2}= \sqrt{25+64}= \sqrt{89}$
• Section Formula
  • Internal Division (ratio $m:n$)
    $P\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right)$
  • External Division (ratio $m:n$)
    $P\left(\frac{mx_2-nx_1}{m-n},\frac{my_2-ny_1}{m-n}\right)$
  • Example: Point dividing $A(2,4)$ and $B(8,10)$ in $3:2$ internally → $P(5,6)$
• Equations of a Straight Line
  • Slope‑Intercept Form – $y = mx + c$
    • $m$ = slope, $c$ = y‑intercept
  • Point‑Slope Form – $y - y_1 = m(x - x_1)$
  • Two‑Point Form – $\displaystyle \frac{y-y_1}{x-x_1}= \frac{y_2-y_1}{x_2-x_1}$
  • General Form – $Ax + By + C = 0$
  • Special Cases
    • Horizontal line: $y = k$ (slope $0$)
    • Vertical line: $x = h$ (undefined slope)
  • Parallel & Perpendicular
    • Parallel: equal slopes ($m_1 = m_2$)
    • Perpendicular: $m_1 \cdot m_2 = -1$
• Midpoint & Segment
  • Midpoint Formula – $M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
  • Application: Finding centre of a line segment, solving geometry problems
• Collinearity
  • Three points $A,B,C$ are collinear if area of $\triangle ABC = 0$
    $\displaystyle \begin{vmatrix} x_1 & y_1 & 1\ x_2 & y_2 & 1\ x_3 & y_3 & 1 \end{vmatrix}=0$
• Area of Polygons
  • Shoelace (Determinant) Formula for triangle $ABC$
    $A = \frac12\big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big|$
  • Extends to quadrilaterals and any simple polygon by adding vertices cyclically
• Centroid, In‑center, Circum‑center
  • Centroid (intersection of medians) – $G\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right)$
  • In‑center – intersection of angle bisectors (equidistant from sides)
  • Circum‑center – intersection of perpendicular bisectors (equidistant from vertices)
• Locus Problems (Intro)
  • Set of points satisfying a condition (e.g., points at a fixed distance from a given point → circle)
  • Typical forms: $x^2 + y^2 = r^2$, $(x-h)^2 + (y-k)^2 = r^2$
• Common Example Problems
  • Find the equation of the line passing through $(2,‑1)$ and parallel to $3x‑4y+5=0$.
  • Determine the distance between points $P(1,2)$ and $Q(4,6)$.
  • Locate the midpoint of the segment joining $A(−3,5)$ and $B(7,‑1)$.
  • Compute the area of triangle with vertices $A(0,0)$, $B(4,0)$, $C(2,3)$ using the shoelace formula.