48  Cartesian Coordinate System

48.1 Introduction

The Cartesian coordinate system is the foundation of Analytical Geometry.
By combining algebra and geometry, it allows us to represent points, lines, curves, and figures using equations.

Every location in the plane is described by an ordered pair \((x,y)\) relative to two perpendicular reference lines:

  • x-axis (horizontal axis)
  • y-axis (vertical axis)
  • Their intersection is the origin \((0,0)\)

This system forms the basis for studying straight lines, circles, conics, vectors, and calculus applications in geometry.

48.2 Quadrants and Signs

The plane is divided into four quadrants:

  1. Quadrant I: \((+,+)\) → \(x>0\), \(y>0\)
  2. Quadrant II: \((-,+)\) → \(x<0\), \(y>0\)
  3. Quadrant III: \((-,-)\) → \(x<0\), \(y<0\)
  4. Quadrant IV: \((+,-)\) → \(x>0\), \(y<0\)

Points lying on axes have one coordinate zero.

48.3 Distance Formula

For points \(A(x_1,y_1)\) and \(B(x_2,y_2)\):

\[ AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]

  • Derived from Pythagoras theorem.
  • Widely used in geometry and coordinate proofs.

Example: \((1,2)\) and \((4,6)\) → \(AB=\sqrt{(3^2+4^2)}=5\).

48.4 Midpoint Formula

Midpoint \(M\) of \(AB\):

\[ M=\Big(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2}\Big) \]

Useful for bisectors, medians of triangles, and symmetry.

48.5 Section Formula

If \(P(x,y)\) divides \(AB\) in ratio \(m:n\) (internally):

\[ P=\Big(\frac{mx_2+nx_1}{m+n},\;\frac{my_2+ny_1}{m+n}\Big) \]

For external division:

\[ P=\Big(\frac{mx_2-nx_1}{m-n},\;\frac{my_2-ny_1}{m-n}\Big) \]

48.6 Collinearity of Points

Three points \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\) are collinear if:

\[ \frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_1}{x_3-x_1} \]

Or equivalently, area of triangle = 0:

\[ \frac{1}{2}\Big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\Big|=0 \]

48.7 Area of a Triangle

For vertices \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\):

\[ \Delta=\frac{1}{2}\Big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\Big| \]

48.8 Slope of a Line

Slope of line through \((x_1,y_1)\), \((x_2,y_2)\):

\[ m=\frac{y_2-y_1}{x_2-x_1},\;\;x_1\neq x_2 \]

  • \(m>0\): line rises
  • \(m<0\): line falls
  • \(m=0\): horizontal line
  • Undefined: vertical line

Parallel lines: equal slopes.
Perpendicular lines: product of slopes = \(-1\).

48.9 Equation of a Line

  • Slope–intercept form: \(y=mx+c\)
  • Point–slope form: \(y-y_1=m(x-x_1)\)
  • Two-point form: \(y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\)
  • Intercept form: \(\frac{x}{a}+\frac{y}{b}=1\)
  • General form: \(Ax+By+C=0\)

48.10 Symmetry of Points

  • Symmetry about x-axis: \((x,y)\to(x,-y)\)
  • Symmetry about y-axis: \((x,y)\to(-x,y)\)
  • Symmetry about origin: \((x,y)\to(-x,-y)\)
  • Symmetry about line \(y=x\): \((x,y)\to(y,x)\)

48.11 Locus of a Point

The locus is the set of all points satisfying a condition.

Examples:
- Equidistant from \((a,0)\) and \((-a,0)\) → locus is \(y\)-axis.
- Equidistant from \((0,a)\) and \((0,-a)\) → locus is \(x\)-axis.
- Distance from origin constant \(r\) → circle \(x^2+y^2=r^2\).

48.12 Equation of a Circle

  • Center \((h,k)\), radius \(r\):

\[ (x-h)^2+(y-k)^2=r^2 \]

  • Center \((0,0)\), radius \(r\):

\[ x^2+y^2=r^2 \]

48.13 Transformation of Axes

  • Translation: Origin shifted to \((h,k)\) → new coordinates:
    \(x'=x-h\), \(y'=y-k\).
  • Rotation: Axes rotated by \(\theta\) →
    \(x=x'\cos\theta-y'\sin\theta\),
    \(y=x'\sin\theta+y'\cos\theta\).

These are useful for simplifying equations.

48.14 Applications

  1. Geometry proofs (collinearity, parallelism, perpendicularity).
  2. Shortest distance between two points or between a point and a line.
  3. Equation of circles, tangents, and normals.
  4. Analyzing symmetry in graphs.
  5. Coordinate methods in mensuration.

48.15 Solved Examples

Example 1
Find distance between \((3,4)\) and \((0,0)\).
\(=\sqrt{3^2+4^2}=5\).

Example 2
Find slope of line joining \((2,3)\) and \((4,7)\).
\(m=\frac{7-3}{4-2}=2\).

Example 3
Equation of line through \((1,2)\) with slope 3.
\(y-2=3(x-1)\Rightarrow y=3x-1\).

Example 4
Equation of circle with center \((2,-1)\) and radius 5.
\((x-2)^2+(y+1)^2=25\).

Example 5
Check if points \((1,2)\), \((2,4)\), \((3,6)\) are collinear.
Slope between \((1,2)\) and \((2,4)\) = 2; slope between \((2,4)\) and \((3,6)\) = 2 → same slope → collinear.

48.16 Practice Problems

  1. Find midpoint of \((2,4)\) and \((6,8)\).
  2. Find slope of line joining \((-1,2)\) and \((3,10)\).
  3. Write equation of line through \((0,5)\) with slope \(-2\).
  4. Find coordinates dividing \((2,3)\) and \((8,9)\) in ratio 1:2 internally.
  5. Find area of triangle with vertices \((0,0)\), \((4,0)\), \((0,3)\).
  6. Write equation of circle with center \((0,0)\) and radius 10.
  7. Find equation of line through \((2,3)\) parallel to \(3x-2y+5=0\).
  8. Find perpendicular distance from \((2,3)\) to line \(4x+3y-10=0\).

48.17 Answer Key

  1. \((4,6)\)
  2. \((10-2)/(3-(-1))=8/4=2\)
  3. \(y-5=-2(x-0)\Rightarrow y=-2x+5\)
  4. \((\frac{1\cdot8+2\cdot2}{3},\frac{1\cdot9+2\cdot3}{3})=(4,5)\)
  5. \(\tfrac{1}{2}|0(0-3)+4(3-0)+0(0-0)|=6\) square units
  6. \(x^2+y^2=100\)
  7. Required line: \(y-3=(3/2)(x-2)\) or \(3x-2y-0=0\)
  8. Distance \(=\frac{|4(2)+3(3)-10|}{\sqrt{4^2+3^2}}=\frac{7}{5}=1.4\)

48.18 Summary

  • Cartesian coordinates describe any point as \((x,y)\).
  • Distance, midpoint, section formulas simplify geometry in algebraic form.
  • Slopes and line equations provide conditions for parallelism and perpendicularity.
  • Circles, locus, and transformations generalize the system to curves.
  • Widely used in higher mathematics, physics, economics, and engineering.