Question

The line \( \frac{x}{5} + \frac{y}{b} = 1 \) passes through the point \( (13, 32) \) and is parallel to the line \( \frac{x}{c} + \frac{y}{3} = 1 \). Then the values of \( b \) and \( c \) are, respectively:

• A. \( -20, \frac{-3}{4} \)
• B. \( 20, \frac{3}{4} \)
• C. \( \frac{3}{4}, 20 \)
• D. \( \frac{-3}{4}, 20 \)
• E. \( -20, \frac{3}{4} \)

Hint:

For parallel lines, equate their slopes and use given points to find unknown parameters.

Answer 1

The Correct Option is A

Since the given lines are parallel, their slopes must be equal. 
Converting the equations into slope-intercept form: \[ \frac{y}{b} = -\frac{x}{5} + 1 \quad \Rightarrow \quad y = -\frac{b}{5} x + b \] \[ \frac{y}{3} = -\frac{x}{c} + 1 \quad \Rightarrow \quad y = -\frac{3}{c} x + 3 \] Equating slopes: \[ -\frac{b}{5} = -\frac{3}{c} \] \[ \frac{b}{5} = \frac{3}{c} \] Cross-multiplying: \[ bc = 15 \] Using the point \( (13, 32) \) in \( \frac{x}{5} + \frac{y}{b} = 1 \): \[ \frac{13}{5} + \frac{32}{b} = 1 \] Solving for \( b \), we find \( b = -20 \). 
Substituting into \( bc = 15 \), we get \( c = \frac{-3}{4} \).