# Cartesian System of Rectangular Co-Ordinates, Intercepts Made by a Line on Axes, Angle Between Two Lines

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## Intercepts Made by a Line on Axes

If a line (not passing through the Origin) meets x-axis at and y-axis at as shown in Fig. then

(i) is called the x-intercept or the intercept made by the line on x-axis.

(ii) is called y-intercept or the intercept made by the line on y-axis.

(iii) and taken together in this order are called the intercepts made by the line on the axes.

(iv) is called the portion of the line intercepted between the axes.

(v) The coordinates of the point on x-axis are and those of point are

To find the intercept of a line in a given plane on x-axis, we put in the given equation of a line and the value of so obtained is called the x intercept.

To find the intercept of a line on y-axis we put and the value of so obtained is called the y intercept.

Example:

If a line is represented by , find its and intercepts.

Solution:

The given equation of the line is

Putting in , we get

Thus, -intercept is .

Again putting in , we get

Thus, -intercept is .

## Angle between Two Lines

Let and be two non-vertical and non-perpendicular lines with slopes and respectively. Let and be the angles subtended by and respectively with the positive direction of x-axis. Then and .

From figure, we have

i.e.

i.e.

As it is clear from the figure that there are two angles and between the lines and .

We know,

If is positive then is positive and is negative i.e. is acute and is obtuse.

If is negative then is negative and is positive i.e. is obtuse and is acute.

Thus the acute angle (say ) between lines and with slopes and respectively is given by

where .

The obtuse angle (say ) can be found by using the formula .

Example:

Find the acute and obtuse angles between the lines whose slopes are and .

Solution:

Let and be the acute and obtuse angle between the lines respectively.

.