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Coordinate Geometry

Intercepts of a Line on the Axes

Science > Mathematics > Coordinate Geometry > Straight Lines > Intercepts of Line on the Axes

In this article, we shall study the concept of intercepts of line on coordinate axes.

If a straight line intercepts X-axis at the point A and Y-axis at point B, then the x-coordinate of A and y-coordinate of B are called x-intercept and y-intercept of the line on the axes respectively.

intercepts of line

Generally, x-intercept is represented by letter ‘a’ and y-intercept is represented by letter ‘b’.

Example 01:

Find slopes and intercepts of the following lines.

  • 3x + 4y = 12

Solution:

Equation of line is 3x + 4y – 12 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (3/4) = – 3/4

To find x-intercept of line substitute y = o in the equation of line

∴  3x + 4 (0) -12 = 0

∴  3x = 12

∴  x = 4

To find y-intercept of line substitute x = 0 in the equation of line

∴  3(0) + 4 y -12 = 0

∴  4y = 12

∴  y = 3

Ans: Slope of line = – 3/4, x-intercept = 4, and y-intercept = 3

  • 2x + 3y = 6

Solution:

Equation of line is 2x + 3y – 6 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (2/3) = – 2/3

To find x-intercept of line substitute y = o in the equation of line

∴  2x + 3(0) – 6 = 0

∴  2x = 6

∴  x = 3

To find y-intercept of line substitute x = 0 in the equation of line

∴  2(0) + 3y – 6 = 0

∴  3y = 6

∴  y = 2

Ans: Slope of line = – 2/3, x-intercept = 3, and y-intercept = 2

  • Y = 3x – 4

Solution:

Equation of line is 3x –y – 4 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (3/-1) = 3

To find x-intercept of line substitute y = o in the equation of line

∴  3x – (0) – 4 = 0

∴  3x = 4

∴  x = 4/3

To find y-intercept of line substitute x = o in the equation of line

∴  3(0) – y -4 = 0

∴  – y = 12

∴  y = – 4

Ans: Slope of line = 3, x-intercept = 4/3, and y-intercept = – 4

  • x/4 – y/3  = 2

Solution:

Equation of line is x/4 – y/3  = 2

Multiplying throughout by 12

∴  3x – 4y = 24

∴  3x – 4y – 24 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (3/-4) = 3/4

To find x-intercept of line substitute y = o in the equation of line

∴  3x – 4(0) – 24 = 0

∴  3x = 24

∴  x = 8

To find the y-intercept of line substitute x = o in the equation of line

∴  3(0) – 4y – 24 = 0

∴  – 4y = 24

∴  y = – 6

Ans: Slope of line = 3/4, x-intercept = 8, and y-intercept = – 6

  • x/2 – y/3  = 1/4

Solution:

Equation of line is x/2 – y/3  = 1/4

Multiplying throughout by 12

∴  6x – 4y = 3

∴  6x – 4y – 3 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (6/-4) = 3/2

To find x-intercept of line substitute y = o in the equation of line

∴  6x – 4(0) – 3 = 0

∴  6x = 3

∴  x = 1/2

To find y-intercept of line substitute x = o in the equation of line

∴  6(0) – 4y – 3 = 0

∴  – 4y = 3

∴  y = – 3/4

Ans: Slope of line = 3/2, x-intercept = 1/2, and y-intercept = – 3/4

  • 2x/3 + y/4  = 5

Solution:

Equation of line is 2x/3 + y/4 = 5

Multiplying throughout by 12

∴  8x + 3y = 60

∴  8x + 3y – 60 = 0

Slope of line = – (coefficient of x/coefficient of y)

∴  Slope of line = – (8/3) = – 8/3

To find x-intercept of line substitute y = o in the equation of line

∴  8x + 3(0) – 60 = 0

∴  8x = 60

∴  x = 60/8 = 15/2

To find y-intercept of line substitute x = o in the equation of line

∴  8(0) + 3y – 60 = 0

∴  3y = 60

∴  y = 20

Ans: Slope of line = – 8/3, x-intercept = 15/2, and y-intercept = 20

Example – 02:

Find the Slope of a line which makes equal intercepts on the axes.

Solution:

intercepts of line

Let line PQ makes an equal intercepts on both the axes.

Thus in right-angled DPOQ, side OP = side OQ

Thus ΔPOQ is right angled isosceles triangle.

Thus m PQO = 45o

Thus line PQ makes an angle of 135o with the positive direction of x-axis

θ = 135o

Slope of line PQ = m = tan θ = tan 135o = -1

Thus the slope of a line making equal intercepts on coordinate axes is -1

Example – 03:

Find the Slope of a line which makes equal intercepts but of opposite signs on the axes.

Solution:

Let line PQ makes an equal but opposite in signs intercepts on both the axes.

Thus in right-angled DPOQ, side OP = side OQ

Thus DPOQ is right angled isosceles triangle.

Thus m PQO = 45o

Thus line PQ makes an angle of 45o with the positive direction of x-axis

θ = 45o

Slope of line PQ = m = tan θ = tan 45o = 1

Thus the slope of a line making equal but opposite in signs intercepts on coordinate axes is 1

Science > Mathematics > Coordinate Geometry > Straight Lines > Intercepts of Line on the Axes

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