Question

The distance between the parallel lines $2x + 5y = 7$ and $2x + 5y = 15$ is ______ (rounded off to 2 decimals).

Hint:

For parallel lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$: Distance = $\dfrac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$.

Answer 1

Correct Answer: 1.47

Step 1: Formula for distance between two parallel lines.
For two lines $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$, \[ \text{Distance} = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \] Step 2: Convert given lines to standard form.
$2x + 5y - 7 = 0$ and $2x + 5y - 15 = 0$
Here, $a = 2$, $b = 5$, $c_1 = -7$, $c_2 = -15$
Step 3: Apply formula.
\[ d = \frac{|-7 - (-15)|}{\sqrt{2^2 + 5^2}} = \frac{8}{\sqrt{29}} = 1.486 \] Step 4: Round off.
\[ d = 1.49 \approx 1.49~\text{units (to 2 decimals)} \] Step 5: Conclusion.
The distance between the two lines is 1.49 units.