Chapter 4: Introduction to Analytic Geometry

Coordinate System & Basic Terms
Quadrants
  • Quadrant I: All points \((x,y)\) with \(x > 0, y > 0\)
  • Quadrant II: All points \((x,y)\) with \(x < 0, y > 0\)
  • Quadrant III: All points \((x,y)\) with \(x < 0, y < 0\)
  • Quadrant IV: All points \((x,y)\) with \(x > 0, y < 0\)
Page: 94
Basic Terms
  • Abscissa: The distance of a point from Y-axis (x-coordinate).
  • Ordinate: The distance of a point from X-axis (y-coordinate).
  • Cartesian Coordinates: The two distances taken together \((x,y)\).
Page: 94
Distance, Section, and Locus Formulas
The Distance Formula
\( d = |AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
The distance between points \(A(x_1, y_1)\) and \(B(x_2, y_2)\). (Page: 95)
Distance of a Point from Origin
\( |OA| = \sqrt{x_1^2 + y_1^2} \)
The distance of point \(A(x_1, y_1)\) from the origin \(O(0,0)\). (Page: 95)
The Ratio Formula (Internal Division)
\( \left(\frac{k_1x_2 + k_2x_1}{k_1 + k_2}, \frac{k_1y_2 + k_2y_1}{k_1 + k_2}\right) \)
Coordinates of a point dividing the line segment AB in the ratio \(k_1:k_2\) internally. (Page: 95)
The Ratio Formula (External Division)
\( \left(\frac{k_1x_2 - k_2x_1}{k_1 - k_2}, \frac{k_1y_2 - k_2y_1}{k_1 - k_2}\right) \)
Coordinates of a point dividing the line segment AB in the ratio \(k_1:k_2\) externally. (Page: 95)
The Midpoint Formula
\( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)
A special case of the ratio formula where \(k_1:k_2 = 1:1\). (Page: 95)
Centroid of a Triangle
\( \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \)
The point of intersection of medians of a triangle with vertices \(A(x_1,y_1)\), \(B(x_2,y_2)\), and \(C(x_3,y_3)\). (Page: 95)
In-centre of a Triangle
\( \left(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c}\right) \)
The point of intersection of angle bisectors. a, b, c are side lengths opposite to vertices A, B, C. (Page: 95)
Area of a Triangular Region
\( \Delta = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \)
Area of a triangle with vertices \(A(x_1,y_1)\), \(B(x_2,y_2)\), and \(C(x_3,y_3)\). (Page: 99)
Transformations of Axes
Translation of Axes
\( X = x - h \)
\( Y = y - k \)
Transforms coordinates \((x,y)\) to \((X,Y)\) when the origin is shifted to \(O'(h,k)\). (Page: 96)
Rotation of Axes
\( X = x \cos\theta + y \sin\theta \)
\( Y = y \cos\theta - x \sin\theta \)
Transforms coordinates \((x,y)\) to \((X,Y)\) when axes are rotated by an angle \(\theta\). (Page: 96)
The Straight Line
Slope and Inclination
Slope/Gradient
\( m = \tan\alpha \)
Where \(\alpha\) is the inclination \((0^\circ < \alpha < 180^\circ)\). (Page: 96)
Slope of a Line Segment
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Slope of the line joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\). (Page: 96)
Equations of Straight Lines
Slope-Intercept Form
\( y = mx + c \)
Line with slope 'm' and y-intercept 'c'. (Page: 97)
Point-Slope Form
\( y - y_1 = m(x - x_1) \)
Line with slope 'm' passing through point \(P(x_1, y_1)\). (Page: 97)
Two-Points Form
\( \frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1} \) or \( \begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0 \)
Line passing through points \(A(x_1, y_1)\) and \(B(x_2, y_2)\). (Page: 97)
Intercepts Form
\( \frac{x}{a} + \frac{y}{b} = 1 \)
Line making x-intercept 'a' and y-intercept 'b'. (Page: 97)
Normal/Perpendicular Form
\( x \cos\alpha + y \sin\alpha = p \)
Line with perpendicular length 'p' from origin, where the perpendicular makes angle \(\alpha\) with the positive X-axis. (Page: 97)
General Form & Parameters
Line: \( ax + by + c = 0 \)
  • Slope: \( m = -\frac{a}{b} \)
  • x-intercept: \( -\frac{c}{a} \)
  • y-intercept: \( -\frac{c}{b} \)
  • Distance from origin: \( p = \frac{|c|}{\sqrt{a^2+b^2}} \)
Page: 97
Family of Lines
\( (a_1x+b_1y+c_1) + h(a_2x+b_2y+c_2) = 0 \)
Represents a family of lines passing through the intersection of lines \(l_1\) and \(l_2\). (Page: 99)
Relationships Between Lines
Conditions for Two Lines
For lines \(l_1: a_1x+b_1y+c_1=0\) and \(l_2: a_2x+b_2y+c_2=0\). (Page: 98)
  • Parallel: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)
  • Perpendicular: \( a_1a_2 + b_1b_2 = 0 \)
  • Identical: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)
  • Intersecting: \( a_1b_2 - a_2b_1 \neq 0 \)
Point of Intersection
\( \left(\frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}, \frac{a_2c_1 - a_1c_2}{a_1b_2 - a_2b_1}\right) \)
The intersection point of two non-parallel lines. (Page: 98)
Condition of Concurrency of Three Lines
\( \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0 \)
The necessary and sufficient condition for three lines to be concurrent. (Page: 98)
Distance of a Point from a Line
\( d = \frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}} \)
Distance from point \(P(x_1, y_1)\) to the line \(ax+by+c=0\). (Page: 98)
Distance Between Two Parallel Lines
\( d = \frac{|c-c'|}{\sqrt{a^2+b^2}} \)
Distance between lines \(ax+by+c=0\) and \(ax+by+c'=0\). (Page: 99)
Angle Between Two Lines
\( \tan\theta = \frac{m_2 - m_1}{1 + m_1m_2} \)
Angle from line \(l_1\) (slope \(m_1\)) to line \(l_2\) (slope \(m_2\)). (Page: 99)
Acute Angle Between Two Lines
\( \tan\theta = \left|\frac{a_1b_2 - a_2b_1}{a_1a_2 + b_1b_2}\right| \)
Acute angle between lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\). (Page: 99)
Pair of Straight Lines
Homogeneous Equation of Second Degree
\( ax^2 + 2hxy + by^2 = 0 \)
Represents a pair of lines through the origin. (Page: 100)
Nature of the Pair of Lines
  • Real and distinct: if \( h^2 > ab \)
  • Real and coincident: if \( h^2 = ab \)
  • Imaginary: if \( h^2 < ab \)
Conditions based on the coefficients of the homogeneous equation. (Page: 100)
Angle Between the Pair of Lines
\( \tan\theta = \frac{2\sqrt{h^2 - ab}}{a+b} \)
Angle \(\theta\) between the lines represented by \(ax^2 + 2hxy + by^2 = 0\). (Page: 100)
Conditions for Angle
  • Coincident Lines (\(\theta=0\)): \( h^2 - ab = 0 \)
  • Perpendicular Lines (\(\theta=\frac{\pi}{2}\)): \( a+b = 0 \)
Special cases for the angle between the pair of lines. (Page: 100)