NCERT Class 11-Math՚s: Exemplar Chapter – 10 Straight Lines Part 1 (For CBSE, ICSE, IAS, NET, NRA 2023)

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10.1 Overview

10.1. 1 Slope of a line

If is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan is called the slope of the line and is denoted by m.

The slope of a line passing through points and is given by

10.1. 2 Angle between two lines:

The angle θ between the two lines having slopes and is given by

If we take the acute angle between two lines, then

If the lines are parallel, then .

If the lines are perpendicular, then .

10.1. 3 Collinearity of three points

If three points and are such that slope of slope of , i.e.. ,

Or then they are said to be collinear.

10.1. 4 Various forms of the equation of a line

(i) If a line is at a distance a and parallel to axis, then the equation of the line is .

(ii) If a line is parallel to y-axis at a distance b from y-axis then its equation is

(iii) Point-slope form: The equation of a line having slope and passing through the point is given by

(iv) Two-point-form: The equation of a line passing through two points and is given by

(v) Slope intercept form: The equation of the line making an intercept c on axis and having slope is given by

Note that the value of will be positive or negative as the intercept is made on the positive or negative side of the y-axis, respectively.

(vi) Intercept form: The equation of the line making intercepts a and b on and axis respectively is given by

(vii) Normal form: Suppose a non-vertical line is known to us with following data:

(a) Length of the perpendicular (normal) p from origin to the line.

(b) Angle which normal makes with the positive direction of x-axis.

Then the equation of such a line is given by

10.1. 5 General equation of a line

Any equation of the form , where A and B are simultaneously not zero, is called the general equation of a line.

Different forms of

The general form of the line can be reduced to various forms as given below:

(i) Slope intercept form: If , then can be written as

If then which is a vertical line whose slope is not defined and x-intercept is .

(ii) Intercept form: If , then can be written as , where .

If , then can be written as which is a line passing through the origin and therefore has zero intercepts on the axes.

(iii) Normal Form: The normal form of the equation is where,

Note: Proper choice of signs is to be made so that should be always positive.

10.1. 6 Distance of a point from a line

The perpendicular distance (or simply distance) of a point from the line is given by

Distance between two parallel lines

The distance between two parallel lines and is given by

10.1. 7 Locus and Equation of Locus:

The curve described by a point which moves under certain given condition is called its locus. To find the locus of a point P whose coordinates are , express the condition involving and . Eliminate variables if any and finally replace by and by to get the locus of P.