An intercept is a point on y-axis, through which the slope of the line passes. It is the value of y-axis, where the line is crossing the axis. This is represented when we write the equation for a line.
Also read: Equation of plane in intercept form
The intercept of a line is the point at which it intersects either the x-axis or y-axis. If the axis is not specified, usually the y-axis is considered. It is normally denoted by the letter ‘b’. E
xcept that line is accurately vertical, it will constantly cross the y-axis somewhere, even if it is way off the top or bottom of the chart.
The equation of the line, which intersects the y-axis at a point is given by:
y = mx + c
Now, we have to write the intercept form of the line, we can replace c with b. Thus, the equation becomes:
y = mx + b
Hence, the formula for the intercept of a line is given by:
b = y – mx
Where, b is the intercept, m is the slope of the line and y and x indicate the points on y-axis and x-axis respectively.
Now, another way of writing the equation of the line, considering a line is intersecting the x-axis and y-axis at point a and b respectively.
x/a + y/b = 1
Here, a and b are the intercepts of the line which intersect the x-axis and y-axis, respectively. The values of a and b can be positive, negative or zero and explain the position of the points at which the line cuts both axes, relative to the origin.
Proof of Intercept Form
The formula of the line formed by the two points is given by:
y-y1/y2-y1 = x-x1/x2-x1
Say, P(a, 0) = (x1, y1) and Q(0, b) = (x2, y2) are the two points of the line which cuts the x-axis and y-axis, relative to the origin(0,0). Then the formula becomes:
=> y – 0 / b – 0 = x – a/ 0 – a
=> y/b = x/-a – a/-a
=> x/a + y/b = 1
Let two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis, respectively. Find the equation of the line.
Solution: Given, two intercepts P(2,0) and Q(0,3) intersect the x-axis and y-axis.
From the equation of the line we know,
x/a + y/b = 1 ……….. (1)
Here, a = 2 and b = 3
Therefore, putting the values of intercepts a and b, in equation 1, we get:
=>x/2 + y/3 = 1
=> 3x + 2y = 6
=> 3x + 2y – 6 = 0, is the equation of the line.