x-intercepts and y-intercepts of a Line – Straight Lines | Class 11 Maths

Line is a straight one-dimensional figure that has no thickness. In geometry, a line extends endlessly in both directions. It is described as the shortest distance between any two points. A-line can also be understood as multiple points connected to each other in one specific direction without a gap between them.

Horizontal Line

Any line which is parallel to X-axis ( or perpendicular to Y-axis ) is called horizontal line.

Examples of horizontal line:

(a) y = 4.3

(b) y = k, where k is any constant

(c) y = 0, this is the equation of X-axis

Slope of a Horizontal Line

The angle between the horizontal line and X-axis is 0°. Thus, slope of any horizontal line = tan(0°) = 0

Slope of Horizontal line = 0

Vertical Line

Any line which is parallel to Y-axis (or perpendicular to X-axis) is called a vertical line.

Examples of vertical line:

(a) x = 10

(b) x = k, where k is any constant

(c) x = 0, this is the equation of Y-axis

Slope of a Vertical Line

The angle between the vertical line and X-axis is 90°. Thus, the slope of any vertical line = tan(90°), which is not defined. 

Slope of vertical line is not defined.

x-Intercept and y-Intercept

x-intercept

The point of intersection of a line and the x-axis is called the x-intercept.

Example:

In figure s is the x-intercept of line AB.

General form of x-intercept is (s, 0)

y-intercept

The point of intersection of a line and the y-axis is called the y-intercept.

Example:

In figure, t is the y-intercept of line AB.

General form of y-intercept is (0, t)

Intercepts from the equation of Line 

Any line, say AB, has the x-intercept as p and y-intercept as q if the equation of the line is of the following form:

(x/p) + (y/q) = 1

Sample Problems on x- and y-intercepts

Problem 1: Find the x and y-intercepts of the line having equation: y = x + 10

Solution:

Converting the equation of the given line in intercept form:

y – x = 10

(y/10) – (x/10) = 1, ———dividing both sides by 10

(y/10) + (-x/10) = 1

(x/(-10)) + (y/10) = 1,

Thus, x-intercept is -10 and y intercept is 10.

Another solution:

x-intercept is of the form (s, 0).

Let us put y = 0 in the equation of the given line:

0 = x + 10 or

x = -10

Thus, x-intercept of given line is -10.

y-intercept is of the form (0, t).

Let us put x = 0 in the equation of the given line:

y = 0 + 10 or

y = 10

Thus, y-intercept of given line is 10.

Problem 2: Find the x and y-intercepts of the line having equation: 20y = 10 – 40x

Solution:

Converting the equation of the given line in intercept form :

20y + 40x = 10

(20y/10) + (40x/10) = 1, ———dividing both sides by 10

(2y/1) + (4x/1) = 1

(x/(1/4)) + (y/(1/2)) = 1,

Thus, x-intercept is (1/4) and y intercept is (1/2).

Another solution:

x-intercept is of the form (s, 0).

Let us put y = 0 in the equation of the given line:

20*(0) = 10 – 40x, or

0 + 40x = 10, or

x = 1/4

Thus, x-intercept of given line is 1/4 or 0.25

y-intercept is of the form (0, t).

Let us put x = 0 in the equation of the given line:

20y = 10 – 40*(0)

20y = 10, or

y = 1/2

Thus, y-intercept of given line is 1/2 or 0.5

Problem 3: Find the x and y-intercepts of the line having equation: 4x + 5y = -3

Solution:

Converting the equation of the given line in intercept form :

4x + 5y = -3 ——-given

4x/(-3) + 5y/(-3) = -3/(-3), —–dividing both sides by -3

x/(-3/4) + y/(-3/5) = 1,

Thus, x-intercept is (-3/4) and y intercept is (-3/5)

Another solution:

x-intercept is of the form (s, 0).

Let us put y = 0 in the equation of the given line:

4x + 5*(0) = -3, or

4x + 0 = -3, or

x = -3/4

Thus, x-intercept of given line is -3/4

y-intercept is of the form (0, t).

Let us put x = 0 in the equation of the given line:

4*(0) + 5y = -3, or

0 + 5y = -3, or

y = -3/5

Thus, y-intercept of given line is -3/5

Problem 4: A line AB has x-intercept = 0. Find its y-intercept.

Solution:

x-intercept of given line is 0.

This means that the point of intersection of the given line and X-axis is (0, 0).

In other words, the given line passes through the origin.

Thus, y-intercept of the given line is 0 (as the point of intersection of the given line and Y-axis is also (0, 0)).

Problem 5: A line passes through the point (3, 4), (p, q), and (c, d), where p and d are x and y-intercepts respectively. Find the value of p, q, c, and d given that the slope of the line is -1/2.

Solution:

p is the x-intercept of the given line, so (p, q) lies on X-axis.

This means that q = 0 ——–(i)

d is the y-intercept of the given line, so (c, d) lies on Y-axis.

This means that c = 0 ——–(ii)

Slope of any line = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are two points that lie on it.

Slope of given line = (4-q)/(3-p), or

-1/2 = (4-0)/(3-p), ——from (i)

(-1)*(3-p) = 4*2, or

p – 3 = 8, or

Thus, p = 11  ——–(iii)

Slope of given line = (4-d)/(3-c), or

-1/2 = (4-d)/(3-0), ——-from (ii)

(-1/2)*3 = 4 – d, or

d – 3/2 = 4, or

d = 4 + 3/2

Thus d = 11/2 or 5.5 ——–(iv)

Thus, the values are : p = 11, q = 0, c = 0, d = 11/2

Intercepts of a line from table

General form of x-intercept is (s, 0), where s is any real number.

Thus, the point which has ordinate 0 is the x-intercept.

General form of x-intercept is (t, 0), where t is any real number.

Thus, the point which has x-coordinate 0 is the y-intercept.

Sample Problems

Keeping the above points in mind, let us solve the following:

Problem 1: Find the x-intercept and y-intercept of the line passing through the following points:

x	8	4	-8	0	-12
y	-5	0	15	5	20

Solution:

The point (4, 0) lies on the x-axis. Thus, x-intercept is 4.

The point (0, 5) lies on the y-axis. Thus, y-intercept is 5.

Problem 2: Find the x-intercept and y-intercept of the line passing through the following points:

x	1	2	-1/2	4
y	1	1/3	2	-1

Solution:

No point in the table is such that it has x-coordinate = 0 or y-coordinate = 0.

So, let us find the equation of the line using two point form :

y – y1 = (y2 – y1)/(x2 – x1) * (x – x1), where (x1, y1) and (x2, y2) are two points.

We can take any two points from the table.

Let us take the points (1, 1) and (4, -1) for ease of calculation. Equation of given line is :

y – 1 = (-1-1)/(4-1) * (x – 1) or

3y – 3 = -2x + 2 or

2x + 3y = 2 + 3 or

2x + 3y = 5,

2x/5 + 3y/5 = 5/5, ——dividing both sides by 5

x/(5/2) + y/(5/3) = 1, which is the equation of given line in intercept form

Thus, x-intercept of given line is 5/2 or 2.5 and y-intercept of given line is 5/3.