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STRAIGHT LINES

10.1 Introduction

I. Distance between the points P (x1, y1) and Q (x2, y2) is

For example, distance between the points (6, – 4) and (3, 0) is

II. The coordinates of a point dividing the line segment joining the points (x1, y1)
and (x2, y2) internally, in the ratio m: n are

For example, the coordinates of the point which divides the line segment joining
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by

III. In particular, if m = n, the coordinates of the mid-point of the line segment
joining the points (x1, y1) and (x2, y2) are

IV. Area of the triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is

For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is

Remark If the area of the triangle ABC is zero, then three points A, B and C lie on
a line, i.e., they are collinear.
10.2 Slope of a Line
The angle (say) made by the line l with positive direction of x-axis and measured anti
clockwise is called the inclination of the line. Obviously 0° 180°
Definition 1 If is the inclination of a line l, then tan is called the slope or gradient of the line l.
The slope of a line is denoted by m. Thus, m = tan , 90°
10.2.1 Slope of a line when coordinates of any two points on the line are given
Let P(x1, y1) and Q(x2, y2) be two points on non-vertical line l whose inclination is .

10.2.2 Conditions for parallelism and perpendicularity of lines in terms of their slopes
If the line l1 is parallel to l2