Key Concepts and Formulas

• Distance Between Parallel Lines: The distance, d, between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.
• Slope-Intercept Form: A line with x-intercept a and y-intercept b has the equation $\frac{x}{a} + \frac{y}{b} = 1$. The slope of this line is $-\frac{b}{a}$.
• Parallel Lines: Two lines are parallel if and only if they have the same slope.

Step-by-Step Solution

Step 1: Find the value of b in the equation of line L

• Why? The equation of line L contains an unknown parameter, b. We can solve for b since we know line L passes through the point (13, 32).
• Action: Substitute the coordinates (13, 32) into the equation $\frac{x}{5} + \frac{y}{b} = 1$. $\frac{13}{5} + \frac{32}{b} = 1$
• Calculation: $\frac{32}{b} = 1 - \frac{13}{5} = \frac{5 - 13}{5} = -\frac{8}{5}$ $b = \frac{32 \cdot 5}{-8} = -4 \cdot 5 = -20$
• Result: The equation of line L is $\frac{x}{5} + \frac{y}{-20} = 1$.

Step 2: Find the value of c in the equation of line K

• Why? The equation of line K contains an unknown parameter, c. We know line K is parallel to line L, which means their slopes are equal. We can use this information to solve for c.
• Action 1: Find the slope of line L
  • Why? We need the slope of L to determine the slope of K, since they are parallel.
  • Calculation: The equation of line L is $\frac{x}{5} + \frac{y}{-20} = 1$. Therefore, the slope of line L is $m_L = -\frac{-20}{5} = 4$.
• Action 2: Equate the slopes of lines L and K
  • Why? Parallel lines have the same slope.
  • Calculation: The equation of line K is $\frac{x}{c} + \frac{y}{3} = 1$. Therefore, the slope of line K is $m_K = -\frac{3}{c}$. Since $L \parallel K$, $m_L = m_K$. $4 = -\frac{3}{c}$ $c = -\frac{3}{4}$
• Result: The equation of line K is $\frac{x}{-3/4} + \frac{y}{3} = 1$.

Step 3: Convert the equations of lines L and K to general form (Ax + By + C = 0)

• Why? The formula for the distance between parallel lines requires the equations to be in general form.
• Action 1: Convert the equation of line L to general form.
  • Equation of L: $\frac{x}{5} + \frac{y}{-20} = 1$
  • Calculation: Multiply both sides by 20: $20 \left(\frac{x}{5} + \frac{y}{-20}\right) = 20(1)$ $4x - y = 20$ $4x - y - 20 = 0$
  • Result: The general form of line L is $4x - y - 20 = 0$.
• Action 2: Convert the equation of line K to general form.
  • Equation of K: $\frac{x}{-3/4} + \frac{y}{3} = 1$ which simplifies to $-\frac{4x}{3} + \frac{y}{3} = 1$
  • Calculation: Multiply both sides by 3: $3\left(-\frac{4x}{3} + \frac{y}{3}\right) = 3(1)$ $-4x + y = 3$ $-4x + y - 3 = 0$
  • Important: To use the distance formula, the coefficients of x and y must be the same. Multiply the equation of line K by -1: $(-1)(-4x + y - 3) = (-1)(0)$ $4x - y + 3 = 0$
  • Result: The general form of line K is $4x - y + 3 = 0$.

Step 4: Calculate the distance between lines L and K

• Why? Now that both equations are in general form, we can apply the distance formula.
• Action: Apply the formula $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.
• Calculation:
  • Line L: $4x - y - 20 = 0$, so $A = 4$, $B = -1$, and $C_1 = -20$.
  • Line K: $4x - y + 3 = 0$, so $A = 4$, $B = -1$, and $C_2 = 3$. $d = \frac{|-20 - 3|}{\sqrt{4^2 + (-1)^2}} = \frac{|-23|}{\sqrt{16 + 1}} = \frac{23}{\sqrt{17}}$

Common Mistakes & Tips

• Sign Errors: Be very careful with signs, especially when substituting values and manipulating equations.
• General Form: Remember that to use the distance formula, the coefficients of x and y must be identical in both equations.
• Slope Calculation: Double-check the slope calculation, especially when using intercept form.

Summary

We found the equations of lines L and K by using the given point and the parallel condition. We then converted these equations to general form and used the distance formula to find the distance between the lines. The distance between the lines is $\frac{23}{\sqrt{17}}$.

The final answer is \boxed{\frac{23}{\sqrt{17}}}, which corresponds to option (C).