Question

The distance between the parallel lines $3x + 4y = 9$ and $6x + 8y = 15$ is

• A. $0.3$ units
• B. $3$ units
• C. $0.5$ units
• D. $5$ units

Hint:

Always ensure your coefficients are matched before running the formula! Alternatively, you could divide the second line equation by 2 to get $3x + 4y - 7.5 = 0$. Then the distance is simply $\frac{|9 - 7.5|}{\sqrt{3^2+4^2}} = \frac{1.5}{5} = 0.3$ units.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the perpendicular distance between two parallel straight lines given by their linear Cartesian equations.

Step 2: Key Formula or Approach:
The shortest distance $d$ between two parallel lines written in the form $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is:
$$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$ Before applying the formula, we must ensure that the coefficients of $x$ and $y$ are completely identical in both equations.

Step 3: Detailed Explanation:
Let's list our two equations:
Line 1: $3x + 4y - 9 = 0$ Line 2: $6x + 8y - 15 = 0$ To make the coefficients match, let's multiply the first equation by 2:
$$2 \cdot (3x + 4y - 9 = 0) \implies 6x + 8y - 18 = 0$$ Now our coefficients are perfectly matched:
$A = 6$
$B = 8$
$C_1 = -18$
$C_2 = -15$
Substitute these parameters into our distance formula:
$$d = \frac{|-18 - (-15)|}{\sqrt{6^2 + 8^2}}$$ $$d = \frac{|-18 + 15|}{\sqrt{36 + 64}} = \frac{|-3|}{\sqrt{100}}$$ $$d = \frac{3}{10} = 0.3 \text{ units}$$ This matches option (A).

Step 4: Final Answer:
The distance between the lines is $0.3$ units, which corresponds to option (A).