Understanding the Equation of a Straight Line

The question asks for the condition under which the equation \( Ax + By + C = 0 \) represents a straight line. This is a fundamental concept in coordinate geometry, dealing with linear equations in two variables, \(x\) and \(y\).

General Form of a Linear Equation

The equation \( Ax + By + C = 0 \) is known as the general form of a linear equation in two variables \(x\) and \(y\). Here, \(A\), \(B\), and \(C\) are real numbers.

Conditions for Representing a Straight Line

Let's consider the different possibilities for the coefficients \(A\) and \(B\):

• Case 1: \(A \neq 0\) and \(B \neq 0\)
  In this case, we can rearrange the equation to the slope-intercept form \( y = mx + c \). If \(B \neq 0\), we can write \( By = -Ax - C \), which gives \( y = -\frac{A}{B}x - \frac{C}{B} \). This is the equation of a straight line with slope \( m = -\frac{A}{B} \) and y-intercept \( c = -\frac{C}{B} \). This clearly represents a straight line.
• Case 2: \(A \neq 0\) and \(B = 0\)
  If \(B = 0\) and \(A \neq 0\), the equation becomes \( Ax + C = 0 \). We can rearrange this to \( Ax = -C \), or \( x = -\frac{C}{A} \). This equation represents a vertical line passing through the point \( (-\frac{C}{A}, 0) \) and parallel to the y-axis. This is also a straight line.
• Case 3: \(A = 0\) and \(B \neq 0\)
  If \(A = 0\) and \(B \neq 0\), the equation becomes \( By + C = 0 \). We can rearrange this to \( By = -C \), or \( y = -\frac{C}{B} \). This equation represents a horizontal line passing through the point \( (0, -\frac{C}{B}) \) and parallel to the x-axis. This is also a straight line.
• Case 4: \(A = 0\) and \(B = 0\)
  If both \(A = 0\) and \(B = 0\), the equation becomes \( 0x + 0y + C = 0 \), which simplifies to \( C = 0 \).
  • If \(C \neq 0\), the equation becomes \( C = 0 \), which is a false statement (e.g., \( 5 = 0 \)). There are no points \( (x, y) \) that satisfy this equation. The solution set is empty, which does not represent a straight line.
  • If \(C = 0\), the equation becomes \( 0 = 0 \), which is a true statement for all values of \(x\) and \(y\). This equation is satisfied by every point \( (x, y) \) in the plane. This represents the entire coordinate plane, not a single straight line.

From the analysis of these cases, we can see that the equation \( Ax + By + C = 0 \) represents a straight line if and only if we are in Case 1, Case 2, or Case 3. These cases cover situations where either \(A \neq 0\) (Case 1 and 2) or \(B \neq 0\) (Case 1 and 3). The only case where it does *not* represent a straight line is when both \(A = 0\) and \(B = 0\). Therefore, the condition for \( Ax + By + C = 0 \) to represent a straight line is that at least one of \(A\) or \(B\) must be non-zero.

Analyzing the Options

Let's evaluate the given options based on our conclusion:

• Option 1: for all real numbers a, b and c.
  This is incorrect because if \(A=0\) and \(B=0\), it's not a line.
• Option 2: only when a ≠ 0.
  This is incorrect because it can be a line even if \(A=0\) (as long as \(B \neq 0\)).
• Option 3: only when b ≠ 0.
  This is incorrect because it can be a line even if \(B=0\) (as long as \(A \neq 0\)).
• Option 4: only when at least one of a and b is non-zero.
  This is correct because it covers all cases where \(A \neq 0\) or \(B \neq 0\) (or both), and correctly excludes the case where both are zero.

Thus, the equation \(Ax + By + C = 0\) represents a straight line if and only if at least one of the coefficients \(A\) and \(B\) is non-zero.

Revision Table: Equation \(Ax + By + C = 0\)

Additional Information: Forms of Straight Line Equations

Besides the general form \( Ax + By + C = 0 \), there are other common forms for the equation of a straight line:

• Slope-Intercept Form: \( y = mx + c \), where \(m\) is the slope and \(c\) is the y-intercept. This form is not suitable for representing vertical lines (where the slope is undefined).
• Point-Slope Form: \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \( (x_1, y_1) \) is a point on the line. This form is also not suitable for vertical lines.
• Intercept Form: \( \frac{x}{a} + \frac{y}{b} = 1 \), where \(a\) is the x-intercept and \(b\) is the y-intercept. This form is not suitable for lines passing through the origin (where \(a=0\) and \(b=0\)), horizontal lines (where \(b\) is undefined unless the line is the x-axis), or vertical lines (where \(a\) is undefined unless the line is the y-axis).

The general form \( Ax + By + C = 0 \) is advantageous because it can represent *any* straight line, including vertical and horizontal lines, for appropriate values of \(A\), \(B\), and \(C\), provided at least one of \(A\) or \(B\) is non-zero.