Question

The distance between the line 3x + 4y + 5 = 0 and 3x + 4y - 7 = 0 is

• A. 2
• B. 3
• C. 5/3
• D. 12/5

Hint:

Before using the parallel distance formula, ensure the $A$ and $B$ coefficients are identical in both equations. If one is a multiple of the other, divide or multiply to make them match.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The two given lines have the same coefficients for $x$ and $y$ (3 and 4), which means they are parallel. The distance between two parallel lines is the constant perpendicular distance between them.

Step 2: Key Formula or Approach:
The distance $d$ between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is: \[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]

Step 3: Detailed Explanation:
Given lines: Line 1: $3x + 4y + 5 = 0 \implies C_1 = 5$ Line 2: $3x + 4y - 7 = 0 \implies C_2 = -7$ Here $A = 3$ and $B = 4$. Applying the formula: \[ d = \frac{|5 - (-7)|}{\sqrt{3^2 + 4^2}} \] \[ d = \frac{|5 + 7|}{\sqrt{9 + 16}} \] \[ d = \frac{12}{\sqrt{25}} \] \[ d = \frac{12}{5} \]

Step 4: Final Answer:
The distance between the lines is 12/5.

Answer 2

Instead of jumping straight to the parallel-line distance formula, the same answer can be reached by picking a specific point on one line and measuring its perpendicular distance to the other, since the two lines never converge, that distance is the same everywhere along them.

\[ \text{Set } y = 0 \text{ in } 3x + 4y + 5 = 0 \implies 3x + 5 = 0 \implies x = -\frac{5}{3} \]

So \(\left(-\dfrac{5}{3}, 0\right)\) is a point on the first line. Its perpendicular distance to the second line \(3x + 4y - 7 = 0\) is:

\[ d = \frac{\left|3\left(-\frac{5}{3}\right) + 4(0) - 7\right|}{\sqrt{3^2+4^2}} = \frac{|-5-7|}{5} = \frac{12}{5} \]

Comparing this to each option below:

1. 2: This would need the numerator of the distance calculation to come out to exactly 10, which doesn't match the \(|-5-7|=12\) obtained from the actual constant terms of these two lines.
2. 3: This would require a numerator of 15, again inconsistent with the difference between the constant terms of the two given line equations.
3. 5/3: This value would arise from mistakenly under-counting the true separation between the constants of the two lines before dividing by \(\sqrt{A^2+B^2}=5\).
4. 12/5: This matches exactly what the point-to-line calculation above produces, using the true point on line 1 and the true coefficients of line 2.

The direct geometric check, placing a real point on one line and measuring how far it sits from the other, confirms the same separation.

Therefore, the correct answer is 12/5.