Question

Find equation of the line which is equidistant from parallel lines 9x+6y-7= 0 and 3x+2y+6=0.
give full explanation with figure..​

Answer 1

3x + 2y + 11/6 = 0

Explanation:-

Given equation:

9 x + 6 y - 7 = 0

3 x + 2 y + 6 = 0

can also be written as,

• 3 x + 2 y - 7/3 = 0
• 3 x + 2 y + 6 = 0

The required equation of line which would be equidistant from these lines must be between the given equation of lines and will be parallel to those lines.

Lines are parallel this implies that lines must have the same slope.

Slope = -a/b for some equation ax + by + c = 0

Therefore the required equation will be of the form, 3x + 2y + k = 0.

Let say,

• 3 x + 2 y - 7/3 = 0 be L1
• 3 x + 2 y + 6 = 0 be L2
• 3 x + 2 y + k = 0 be L3

Distance between two parallel lines is given by,

• |c1 - c2|/√(a²+b²)

Distance b/w L1 & L3 = Distance between L2 & L3

=> |k + 7/3|/√(3²+2²) = |k - 6|/√(3²+2²)

=> |k + 7/3/√(9 + 4) = |k - 6|/√(9 + 4)

=> |k + 7/3| = |k - 6|

Squaring both sides

=> k² + 49/9 + 14k/3 = k² + 36 - 12k

=> 14k/3 + 12k = 36 - 49/9

=> 50k/3 = 275/9

=> k = 275/9 × 3/50

=> k = 11/6

So the required equation of lines will be 3x + 2y + 11/6 = 0

Answer 2

Answer:

Equation of the line which is equidistant from given parallel lines is 18x+12y + 11 = 0