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Intercept is a point on the y-axis of a graph through which the slope of a line passes. It is defined as the y-coordinate of a point where a line or curve intersects the y-axis.
- The equation of a line is represented as y = mx+c, where m is the slope and c is the y-intercept.
- There are basically two intercepts namely the x-intercept, and the y-intercept.
- X-Intercept is the point where the line intersects the x-axis on a graph.
- The point where the line crosses the y-axis is referred to as the Y-Intercept.
- If a line has an intercept on either of the axes, the other coordinate of that point will be zero.
The equation of a line can be determined either by knowing only its slope and one intercept or by knowing its both intercepts.
Read More: NCERT Solutions For Class 11 Maths Straight Lines
Key Terms: Intercept, X-Intercept, Y-Intercept, Coordinates, Slope Intercept Form, Intercept Formula, Equation of a Line, Straight Line
Intercept Meaning
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Intercept is the point where a straight line or a curve intersects the axis of the graph.
- ‘X-axis’ (Horizontal) and the ‘Y-axis’ (Vertical) are the two perpendicular axes in the cartesian coordinate system.
- Intercept is defined as a point that intersects either the x-axis or y-axis.
- If a point intersects the x-axis, then it is referred to as the x-intercept.
- If a point intersects the y-axis, then it is referred to as the y-intercept.
- If the axis is not specified, then it generally represents the y-intercept.
- Intercept is generally denoted by the letter ‘b’.
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Intercept Formula
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The general equation of a line that intersects the y-axis at a point is given as
| y = mx + c |
As we need to write the intercept form of the line, we can replace c with b. Therefore, the equation will be
y = mx + b
Thus, the Intercept Formula for the y-axis is given as
| b = y – mx |
Where
- b denotes the intercept.
- m is the slope of the line.
- x and y are the points on the x-axis and y-axis respectively.
If a line is intersecting the x-axis and y-axis at points a and b respectively, then the Intercept Formula is written as
| \({x \over a} + {y \over b} = 1\) |
Where
- a is the x-intercept, a point that intersects the x-axis.
- b is the y-intercept, a point that intersects the y-axis.
Read More: Straight Lines Important Questions
How to Find X and Y Intercepts?
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Consider that equation of a straight line is given as Ax + By = C.
Now, on dividing the equation by C, we get
Ax/C + By/C = C/C
[x/(C/A)] + [y/(C/B)] = 1
If we compare this equation with the equation of a line in intercept form (x/a) + (y/b) = 1, we get
- a = x-intercept = C/A
- b = y-intercept = B/C
X-Intercept
In order to find the x-intercept, we will substitute y = 0 and solve for x.
Ax + B(0) = C
Ax = C
x = C/A
Y-Intercept
In order to find the y-intercept, we will substitute x =0 and solve for y.
A(0) + By = C
By = C
y = C/B
Solved ExampleExample: The equation of a straight line is given as 5x +2y =10. Find the x-intercept and the y-intercept. Solution: The straight-line equation is given as 5x +2y =10. Thus, To find the x-intercept, substitute y = 0 in the given equation. 5x + 2(0) = 10 5x =10 x =2 To find the y-intercept, substitute x = 0 in the given equation. 5(0) + 2y =10 2y = 10 y = 5 Thus, the x-intercept is (2, 0) and the y-intercept is (0, 5). |
Two Point Form
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When the x and y-intercepts of a line are given, we use the intercept form to write the equation of that line. The formula of the line formed by the two points is given as
| y-y1/y2-y1 = x-x1/x2-x1 |
Proof of Two Point Form
Consider that P (a, 0) = (x1, y1) and Q (0, b) = (x2, y2) are the two points of a line that intersects the x-axis and y-axis with respect to the origin (0,0).
Thus, the formula will be:
y – 0/b – 0 = x – a/0 – a
y/b = x/-a – a/-a
x/a + y/b = 1
Hence Proved.
Slope-Intercept Form
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When a line is known to us only by its slope and one of its intercepts, we use the slope-intercept form to find its equation. If there is a line with slope ‘m’ and y-intercept ‘c’, then its equation in the slope-intercept form will be
y = mx + c
Derivation of Slope Intercept Form
Consider a line whose slope is 'm' that intersects the y-axis at (0, b). Thus, the y-intercept is b. Consider an arbitrary point (x, y) on the given line.
Assume that (x1, y1) = (0, b) and (x2, y2) = (x, y).
The slope of a line joining two points (x1, y1) and (x2, y2) is given as
| m = (y2 - y1)/(x2 - x1) |
Thus, the slope of the given line will be
m = (y - b) / (x - 0)
m = (y - b) / (x)
Multiplying both sides of the equation by x, we get
mx = y - b
Now, on adding 'b' on both sides,
| y = mx + b |
Thus, the general equation of a line is derived and is referred to as the slope-intercept form.
Read More: Slope Intercept Form Formula
Intercept Graph
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Intercepts are the points on a graph at which the graph crosses the two axes i.e., the x-axis and the y-axis.
- The point where the graph crosses the x-axis is the x-coordinate.
- The point where the graph crosses the y-axis is the y-coordinate.
The equation of the line-making intercepts a and b on the x-and y-axis, respectively is x/a + y/b = 1.
Read More: Applications of Linear Graphs
Solved Examples on Intercept Formula
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Here are a few solved examples on intercept formula and related concepts:
Example 1: A line passes through a point P (1, 2) in such a manner that its intercept between the axes is bisected at P. Find the equation of the line.
Solution: The equation of a line with intercepts a and b at the x-axis and y-axis respectively will be given as
x/a + y/b = 1
The point is given as (1, 2). Thus,
1 = (a + 0)/2
So, a = 2
2 = (0 + b)/2
So, b = 4
Thus, the equation of the given line is
x/2 + y/4 = 1
2x + y – 4 = 0
Example 2: Determine x and y-intercepts of the line represented by the equation 3x + 4y = 12.
Solution: The equation of the line is given as 3x + 4y = 12.
In order to find the y-intercept of the line, the value of x will be taken as 0 in the equation for the line.
3x + 4y = 12
3(0) + 4y = 12
4y = 12
y = 12/4 = 3
To find the x-intercept, the value of the y will be 0 in the equation for the line.
3x + 4y = 12
3x + 4(0) = 12
3x = 12
x = 12/3 = 4
Thus, the x and y-intercepts of the line are calculated as 4 and 3 respectively.
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Things to Remember
- Intercept is the point where the line or curve intersects the axis of the graph.
- If a point intersects the x-axis, then it is the x-intercept and if it intersects the y-axis, then it is the y-intercept.
- If there is a line with slope ‘m’ and y-intercept ‘c’, then the slope-intercept form will be y = mx + c.
- If a line with the x-intercept ‘a’ and y-intercept ‘b’ is given, the intercept formula is x/a + y/b = 1.
- A line formed by two points is given by the formula y-y1/y2-y1 = x-x1/x2-x1.
- The points where the line intersects the x-axis and y-axis are the x-coordinate and y-coordinate respectively.
Previous Years’ Questions
- A straight line makes an intercept on the Y-axis twice as long as that… (TS EAMCET - 2017)
- If a line intercepted between the coordinate axes is trisected at a… (JEE Main - 2014)
- A plane makes intercepts a, b, c at A, B, C on the coordinate axes… (KEAM)
- (0,−1) and (0, 3) are two opposite vertices of a square. The other… (BITSAT - 2006)
- The length of the straight line x - 3y = 1 intercepted by… (VITEEE - 2007)
- If the x-intercept of some line L is double that of the line… (JEE Main - 2013)
- If m is the slope of one of the lines represented by… (KCET - 2010)
- The locus of the point of intersection of lines…
- A straight line through the point A(3, 4) is such that its intercept… (VITEEE - 2012)
- Variable straight lines y = mx + c make intercepts on the curve… (AP EAMCET - 2019)
Sample Questions
Ques. Let two intercepts P (2,0) and Q (0,3) intersect the x-axis and y-axis, respectively. What is the equation of the line? (3 Marks)
Ans. As given, two intercepts P (2,0) and Q (0,3) intersect the x-axis and y-axis.
From the equation of the line we have,
x/a + y/b = 1 ……….. (1)
Here, a = 2 and b = 3
Therefore, by 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
Hence, the equation of the line is 3x + 2y – 6 = 0.
Ques. What is Intercept in Maths? (2 Marks)
Ans. Intercept in Maths is defined as a point where the straight line or a curve intersects the x-axis or the y-axis in a plane. if the line intersects the x-xis, it will be the x-intercept or if the line intersects the y-axis, it will be the y-intercept. When the axis is not mentioned, it is also said to be a y-intercept.
Ques. Find the equation of the line, which makes intercepts –3 and 2 on the x- and y-axes respectively. (3 Marks)
Ans. Given,
- a = – 3
- b = 2
By the intercept form, we know that;
x/a + y/b = 1
x/-3 + y/2 = 1
Or, 2x – 3y + 6 = 0.
Therefore, this is the required equation.
Ques. Find the equation of a line that cuts off equal intercepts on the coordinate axes and passes through the point (2, 3). (3 Marks)
Ans. The intercept form of a line is x/a + y/b = 1.
Because the line has equal intercepts on the coordinate axes, thus,
a = b
So, we get
x/a + y/a = 1
or, x + y = a …(i)
Since, the line passes through points (2, 3), thus, 2 and 3 will satisfy equation (i). Now, putting x = 2 and y = 3 in equation (i), we get
a = 2 + 3 = 5
Putting the value of ‘a’ in equation ( i ), we get-
x + y = 5
or, x + y – 5 = 0 …(ii)
Thus, x + y -5 = 0 is the required equation.
Ques. What is the Intercept Formula? (2 Marks)
Ans. The intercept formula for y-intercept is given as
b = y – mx
Here
- b is the y-intercept.
- m is the slope of the line.
Ques. Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9. (3 Marks)
Ans. Let, the x-intercept of the given line be ‘a’, then the y-intercept will be ‘9 – a’ and the equation of the line will be-
x/a + y/(9 – a) = 1
or, x(9 – a) + ay = a(9 – a) …(i)
Now putting x = 2 and y = 2 in equation (i), we get-
2(9 – a) + 2a = a(9 – a)
or, 18 – 2a + 2a = 9a – a2
or, a2 – 9a + 18 = 0
or, a2 – 3a – 6a + 18 = 0
or, a(a – 3) – 6(a – 3) = 0
or, (a – 3)(a – 6) = 0
= > a = 3 or a = 6
Now, putting the value of a = 3 in equation (i), we get
x(9 – 3) + 3y = 3(9 – 3)
or, 6x + 3y = 18
or, 6x + 3y – 18 = 0 …(ii)
Again putting the value of a = 6 in equation (i), we get-
x(9 – 6) + 6y = 6(9 – 6)
or, 3x + 6y = 18
or, 3x + 6y – 18 = 0 …(iii)
Thus, 6x + 3y – 18 = 0 and 3x + 6y -18 = 0 are the required equations of the given line.
Ques. Reduce 6x + 3y – 5 = 0 into slope-intercept form and find its slope and y-intercept. (3 Marks)
Ans. We can write the given equation as
6x + 3y = 5, or
3y = -6x + 5, or
y = -6x/3 + 5/3, or,
y = -2x + 5/3 …(i)
So, y = -2x + 5/3 is the slope-intercept form of the line 6x + 3y – 5 = 0.
Comparing equation (i) with y = mx + c, we get m = -2 and c = 5/3, which implies that the slope of the given line is -2 and its y-intercept is 5/3.
Ques. What will be the value of the x-coordinate if a line intersects the y-axis at a point? (1 Mark)
Ans. The value of the x-coordinate will be zero if a line intersects the y-axis at a point.
Ques. Reduce 4x – 3y = 6 into intercept form and find its intercepts on the axes. (3 Marks)
Ans. We can write the given equation as
4x + (-3)y = 6, or,
4x/6 + (-3)y/6 = 1, or,
2x/3 + (-1)y/2 = 1, or,
x/(3/2) + y/(-2) = 1 …(i)
So, x/(3/2) + y/(-2) = 1 is the intercept form of the line 4x – 3y = 6.
Comparing equation (i) with x/a + y/b = 1, we get a = 3/2 and b = -2, which implies that 3/2 and -2 are the x and y-intercepts of the given line.
Ques. Find the equation of the line perpendicular to the line x – 7y + 5 = 0 and having x-intercept 3. (3 Marks)
Ans. Given the x-intercept of the second line is d1 = 3. Let m1 and m2 be the slopes of given lines.
We can write the given equation of first-line as:
-7y = -x – 5, or,
y = x/7 + 5/7
It means the slope of the first line is m1 = 1/7.
Since both the lines are perpendicular, the product of their slopes will be -1,i.e.
m1m2 = -1, or,
(1/7)m2 = -1, or,
m2 = -7
Thus, the slope of the second line is -7.
Since the equation of the line having slope ‘m’ and x-intercept ‘d’ is y = m(x – d).
The equation of the second line will be
y = m2(x – d1), or,
y = -7(x – 3), or,
y = -7x + 21
Thus the equation of the line perpendicular to x – 7y + 5 = 0 and having x-intercept 3 is y = -7x + 21.
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