Find the distance between two parallel lines.

Understand the Problem

The question is asking for a mathematical calculation of the distance between two parallel lines. To solve this, we use the formula for the distance between two parallel lines, which depends on their equations in the form Ax + By + C1 = 0 and Ax + By + C2 = 0.

Answer

$d = \frac{3}{\sqrt{13}}$

Steps to Solve

Identify the equations of the lines

We need to express both parallel lines in the form $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$. For example, let's say we have two parallel lines given by $2x + 3y - 6 = 0$ and $2x + 3y - 9 = 0$. Here, $A = 2$, $B = 3$, $C_1 = -6$, and $C_2 = -9$.

Use the distance formula

The formula to find the distance $d$ between the two parallel lines is given by: $$ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} $$

Substitute the values into the formula

Using our identified values, we substitute $C_1 = -6$, $C_2 = -9$, $A = 2$, and $B = 3$ into the distance formula: $$ d = \frac{|-9 - (-6)|}{\sqrt{2^2 + 3^2}} $$

Calculate the numerator

Perform the subtraction in the numerator: $$ -9 - (-6) = -9 + 6 = -3 $$ Thus, the absolute value is: $$ |C_2 - C_1| = |-3| = 3 $$

Calculate the denominator

Now calculate $A^2 + B^2$: $$ A^2 + B^2 = 2^2 + 3^2 = 4 + 9 = 13 $$ So, $$ \sqrt{A^2 + B^2} = \sqrt{13} $$

Find the final distance

Now substitute the values back into the distance formula: $$ d = \frac{3}{\sqrt{13}} $$

The distance between the two parallel lines is $d = \frac{3}{\sqrt{13}}$.

More Information

To express the answer in decimal form, you can approximate it as: $$ d \approx 0.832 $$ This represents the shortest distance between the pair of parallel lines in the coordinate plane.

Tips

Confusing the coefficients $A$ and $B$ when identifying the line equations.
Forgetting to take the absolute value of the difference $C_2 - C_1$.
Not simplifying the expression properly, leading to incorrect decimal approximations.

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