Question

The distance between the parallel lines $5x + 12y - 3 = 0$ and $5x + 12y + 10 = 0$ is:

• A. $1$
• B. $2$
• C. $\frac{13}{17}$
• D. $\frac{7}{13}$

Hint:

Since $(5, 12, 13)$ is a standard Pythagorean triplet, the denominator $\sqrt{5^2+12^2}$ is instantly $13$. The difference in constants is $10 - (-3) = 13$. The distance is simply $13/13 = 1$.

Answer 1

The Correct Option is A

Step 1: Distance between parallel lines.
For two parallel lines with the same $A$ and $B$, the gap is $\frac{|C_1 - C_2|}{\sqrt{A^2+B^2}}$.

Step 2: Read the constants.
Here $A = 5$, $B = 12$, $C_1 = -3$, $C_2 = 10$.

Step 3: Find the top.
\[ |C_1 - C_2| = |-3 - 10| = 13 \]

Step 4: Find the bottom.
\[ \sqrt{5^2 + 12^2} = \sqrt{169} = 13 \]

Step 5: Divide.
\[ d = \frac{13}{13} \]

Step 6: Answer.
\[ d = 1 \] \[ \boxed{ 1 } \]