Question

Find combined equation of pair of lines passing through (-1,2), one is parallel to (x+3y-1=0) and th other is perpendicular to (2x-3y-1=0)

Answer 1

common point (-1,2)

for the parallel line:

we know that the slope will be the same so,

x + 3y - 1 = 0

or one can also write it as,

$y = \frac{x}{3} - \frac{1}{3}$

so the slope will be 1/3

equation,

y = x/3 + c

to find the y-intercept we must substitute the values,

2 = -1/3 + c

6 = -1 + 3c

3c = 6 + 1

3c = 7

c = 7/3

equation 1:

$y = \frac{x}{3} +\frac{7}{3}$

or one can also write it as,

$x - 3y + 7 = 0$

for the perpendicular line:

we know that the slope will be the negative reciprocal so,

2x - 3y - 1 = 0

or one can also write it as,

$y = \frac{2}{3} x - \frac{1}{3}$

so the slope will be -3/2

equation

y = -3x/2 + c

to find the y-intercept we must substitute the values,

2 = 3/2 + c

4 = 3 + 2c

2c = 4 - 3

2c = 1

c = 1/2

equation 2:

$y = -\frac{3}{2} x + \frac{1}{2}$

or one can also write it as,

$-3x + 2y + 1 = 0$