Four friends — Ranju, Meena, Farhan, and Toshi — are solving problems.
Ranju: I noticed something! If c = 0 in the standard form of a line ax + by + c = 0, then the line must pass through the origin. Look, if I substitute x = 0 and y = 0, in equation ax + by = 0, the equation is satisfied. So, the origin lies on the line! We can also say that the line passes through the origin.
Farhan: Let us try for the equation 2x + 3y = 0. Here a = 2, b = 3 but c = 0. If we substitute x = 0, then we get 3y = 0 or y = 0. This means (0, 0) lies on the line. So yes, this line passes through the origin.
Toshi: Suppose our equation has b = c = 0, say, 5x = 0. This becomes x = 0 which is the equation of the y-axis. And the y-axis passes through the origin.
Meena: And if we take a = c = 0, say, 7y = 0, that means y = 0, which is the equation of the x-axis that also passes through the origin.
So, they conclude: Whenever c = 0, the line ax + by + c = 0 will always pass through the origin, irrespective of the values of a or b. Do you agree with them?
Step-by-step solution
Idea: A point lies on a line exactly when its coordinates satisfy the equation. Test the origin (0, 0) in ax + by = 0 for general a and b.
- With c = 0 the line is ax + by = 0, where a and b are not both zero.½ mark
- Substitute x = 0, y = 0: LHS = a × 0 + b × 0 = 0 = RHS. This is true for every value of a and b, so the origin always lies on the line.½ mark
- The friends’ special cases fit: 2x + 3y = 0 (a slanting line through O), 5x = 0 (the y-axis), 7y = 0 (the x-axis). So yes, we agree.½ mark
- The reverse is also true: if (0, 0) is on ax + by + c = 0, then 0 + 0 + c = 0, so c = 0. A line passes through the origin exactly when c = 0.½ mark
Check: Try −4x + 9y = 0: (0, 0) gives 0 = 0 ✓. Try x + y = 1 (c ≠ 0): (0, 0) gives 0 ≠ 1, so it misses the origin, as expected.
Answer to write in the exam
c = 0 ⇒ line: ax + by = 0
Put (0, 0): LHS = a(0) + b(0) = 0 = RHS, for all a, b
Conversely, (0, 0) on ax + by + c = 0 ⇒ c = 0
∴ (0, 0) lies on the line; yes, the line always passes through the origin
Common mistakes that cost marks
- Checking only one example, like 2x + 3y = 0, and calling that a proof. The general substitution with letters a and b is what proves it.
- Thinking the rule fails when a or b is 0. Those cases give the axes, which also pass through the origin.
How this can come in the exam
The line px + 4y + (p − 3) = 0 passes through the origin when p =
- 0
- 3
- −3
- 4
Show answer
(B) 3
Through (0, 0) ⇒ constant term = 0 ⇒ p − 3 = 0 ⇒ p = 3.
Try one yourself
Which of these lines pass through the origin: y = 3x, x + y = 2, 4x = 7y?
Show answer
y = 3x (3x − y = 0) and 4x = 7y (4x − 7y = 0): both have c = 0. x + y = 2 does not.
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