Find equation of the line which is equidistant from parallel lines and .
18x + 12y + 11 = 0
Explanation
The question asks for the equation of a line equidistant from two parallel lines. The key concept from the textbook is that for parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0, the line equidistant from both has the equation Ax + By + (C₁ + C₂)/2 = 0. First, we need to make the coefficients of x and y identical in both equations by multiplying the second equation by 3. Then applying the formula gives us the required line.
Solution Steps
Step 1: Check if lines are parallel. Line 1: 9x + 6y - 7 = 0 has slope = -9/6 = -3/2. Line 2: 3x + 2y + 6 = 0 has slope = -3/2. Since slopes are equal, lines are parallel.
Step 2: Make coefficients identical. Multiply Line 2 by 3: 9x + 6y + 18 = 0. Now both lines have same A = 9, B = 6.
Step 3: Apply formula for equidistant line. For lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0, equidistant line is Ax + By + (C₁ + C₂)/2 = 0. Here C₁ = -7, C₂ = 18.
Step 4: Calculate the constant term. (C₁ + C₂)/2 = (-7 + 18)/2 = 11/2
Step 5: Write the equation. 9x + 6y + 11/2 = 0. Multiplying by 2: 18x + 12y + 11 = 0