Line (in 2 Dimensions)

A (straight) line is the simplest of geometrical figures. Imagining a line is very easy. The condition of straightness is also very intuitive. We now formally define a line as a locus of a point satisfying a condition.

NOTE : Throughout this chapter, we will be using the coordinate geometry.

  • Inclination of a Line

The smallest non-negative angle made by a line with positive {X} axis is known as the inclination.  It is generally denoted by {\theta}.

I) Parallel lines have same inclinations.

II) Inclination of {X} axis is {0^o} and inclination of {Y} axis is {90^o}.

  • Slope or Gradient of a Line , {m}

If {\theta} is the inclination, then slope is given by {tan (\theta)}.

NOTE : Knowing trigonometric ratios of particular angles is beneficial.

I) Parallel lines have same slope.

II) Slope of {X} axis is {0} and slope of {Y} axis is not defined.

III) Product of slopes of perpendicular lines is {-1}. This property is extremely important.

If {(x_1, y_1)} and {(x_2, y_2)} are 2 distinct points on a line, its slope is given by

{m = \frac {y_2 - y_1}{x_2 - x_1}}

  • Formal Definition

Line is locus of points, such that slope of line segment joining any 2 distinct points on the line is same.

NOTE : Since line is a locus of points, it has an equation.

  • Intercepts

The {X} and {Y} intercepts of a line are the {X} and {Y} coordinates of the points, where the line meets respective axes. These being directed distances, can be either positive or negative.

  • Equations of Line in Various Forms

I) Slope- Point Form

Equation of a line passing through point {(x_1,y_1)} and having a slope {m} is

{y- y_1 = m (x-x_1)}

II) Two-Point Form

Equation of a line passing through points {(x_1,y_1)} and {(x_2,y_2)} is given by

{\frac {y-y_1}{y_2 - y_1} = \frac {x - x_1}{x_2 - x_1}}

III) Slope-Intercept Form

Equation of a line having slope {m} and {y} intercept {c} is given by

{y = mx +c}

IV) Double-Intercept Form

Equation of a line having $X$ and $Y$ intercepts as $a$ and $b$ is given by

{\frac {x}{a} + \frac {y}{b} = 1}

V) Normal Form (Rarely Used)

If {p} is the length of perpendicular from origin to the line and {\alpha} is the inclination of the perpendicular segment, then

{x \ cos (\alpha) + y \ sin (\alpha) = p}

VI) Parametric Form (More Effective in 3D)

If {\theta} is the inclination and {(x_1,y_1)} is the point on a line, then the parametric equations are given by

{x = x_1 + r \ cos (\theta) \ and \ y = y_1 + r \ sin (\theta),}

where {r} is the distance of {(x,y)} from {(x_1, y_1)}

  • General Equation of 1st Order in 2 Variables

The general equation of 1st order in 2 variables is of the form

{ax + by + c = 0}

It represents a straight line.

The necessary condition is both {a} and {b} cannot be simultaneously zero.

  • Angle Between Any 2 Lines

Let {\theta} be the angle between 2 lines having slopes {m_1} and {m_2}. Then,

{tan (\theta) = \begin {vmatrix} \frac {m_1 - m_2}{1+ m_1 m_2} \end {vmatrix}}

  • Point of Intersection of 2 Lines

We know that 2 non-parallel lines intersect each other at only 1 point. Let the equations of lines be

{a_1 x + b_1 y = c_1 \ and \ a_2 x + b_2 y = c_2}

The coordinates of point of intersection are

{\begin {pmatrix}\frac {\begin {vmatrix} c_1 & c_2 \\ b_1 & b_2 \end {vmatrix}}{\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix}} , \frac {\begin {vmatrix} a_1 & a_2 \\ c_1 & c_2 \end {vmatrix}}{\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix}} \end {pmatrix}}

The condition is,

{\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix} \ne 0}

  • Condition for Concurrency of 3 Lines

Lines are said to be concurrent, when all of them intersect at a point.

Let the lines be {a_1 x + b_1 y + c_1 = 0} , {a_2x + b_2 y + c_2 = 0} and {a_3x + b_3 y + c_3 = 0} . The lines are concurrent, when

{\begin {vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end {vmatrix} = 0}

The point of concurrence can then be obtained easily by considering any 2 of those 3 lines.

  • Length of Perpendicular from a Point to a Line

Let {P (x_1,y_1)} be a point and {ax+by+c=0} be a line. Then, the length of perpendicular from {P} to the line is given by

{\begin {vmatrix} \frac {ax_1 + by_1 + c}{\sqrt {a^2+b^2}} \end {vmatrix}}

  • Distance between 2 Parallel Lines

We know that parallel lines have same slope. This means, in their equations, the ratio of coefficients of {x} and {y} will be same. Let the lines be {ax+by+c_1 = 0} and {ax+by+c_2 = 0}. The distance is given by

{\begin {vmatrix} \frac {c_1 - c_2}{\sqrt {a^2+b^2}} \end {vmatrix}}

  • Family of Straight Lines

This is a useful concept. We know that, through a single point, infinitely many lines pass. Let {u \equiv a_1x + b_1 y + c_1 =0} and {v \equiv a_2 x + b_2 y + c_2 =0} be 2 intersecting lines. Then, equation of family of lines passing through the point of intersection is given by

{u + \lambda v = 0, \ \lambda \in \mathbb R,}

where {\lambda} is a parameter.

Posted in XI

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