Question

Which of the following equations represents a line passing through the origin :

• A. \(2x+3=0\)
• B. \(3y+2=0\)
• C. \(2x+3y+5=0\)
• D. \(2x+3y=0\)

Hint:

A quick way to check if a line passes through the origin \((0,0)\): 1. Look at the equation of the line (usually in the form \(Ax + By + C = 0\) or \(y = mx + c\)). 2. If the equation has {no constant term} (or the constant term \(C\) is 0), then the line passes through the origin. In \(Ax + By + C = 0\), if \(C=0\), it passes through the origin. In \(y = mx + c\), if \(c=0\) (the y-intercept is 0), it passes through the origin. Option (1): \(2x+0y+3=0 \Rightarrow C=3 \neq 0\). Option (2): \(0x+3y+2=0 \Rightarrow C=2 \neq 0\). Option (3): \(2x+3y+5=0 \Rightarrow C=5 \neq 0\). Option (4): \(2x+3y+0=0 \Rightarrow C=0\). This one passes through the origin.

Answer 1

The Correct Option is D

Concept: A line passes through the origin if the coordinates of the origin, \((0,0)\), satisfy the equation of the line. This means if we substitute \(x=0\) and \(y=0\) into the equation, the equation should hold true (e.g., result in \(0=0\)). Alternatively, an equation of a line in the form \(Ax + By + C = 0\) passes through the origin if and only if the constant term \(C\) is zero. Step 1: Understanding the condition for a line to pass through the origin The origin has coordinates \((x, y) = (0, 0)\). If a line passes through the origin, then substituting \(x=0\) and \(y=0\) into its equation must result in a true statement. Step 2: Check each option by substituting \(x=0\) and \(y=0\)
Option (1): \(2x+3=0\) Substitute \(x=0\): \(2(0) + 3 = 0 \Rightarrow 0 + 3 = 0 \Rightarrow 3 = 0\). This is false. So, this line does not pass through the origin. (This line is \(x = -3/2\), a vertical line).
Option (2): \(3y+2=0\) Substitute \(y=0\): \(3(0) + 2 = 0 \Rightarrow 0 + 2 = 0 \Rightarrow 2 = 0\). This is false. So, this line does not pass through the origin. (This line is \(y = -2/3\), a horizontal line).
Option (3): \(2x+3y+5=0\) Substitute \(x=0\) and \(y=0\): \(2(0) + 3(0) + 5 = 0 \Rightarrow 0 + 0 + 5 = 0 \Rightarrow 5 = 0\). This is false. So, this line does not pass through the origin.
Option (4): \(2x+3y=0\) Substitute \(x=0\) and \(y=0\): \(2(0) + 3(0) = 0 \Rightarrow 0 + 0 = 0 \Rightarrow 0 = 0\). This is true. So, this line passes through the origin. Step 3: Conclusion The equation \(2x+3y=0\) represents a line passing through the origin because its constant term is zero, and substituting \((0,0)\) satisfies the equation.