Question

Find the equation of the lines parallel and perpendicular to the line 5x+2y=12 through the point (-2,3)

Answer 1

The equation of line parallel to given line and passing through points ( - 2 , 3 ) is 5 x + 2 y + 4 = 0

The equation of line perpendicular to given line and passing through points ( - 2 , 3 ) is 2 x - 5 y + 19 = 0

Explanation:

Given equation of line as :

5 x + 2 y = 12

or, 2 y = - 5 x + 12

or , y =
`(-5)/(2)` x +
`(12)/(2)`

Or, y =
`(-5)/(2)` x + 6

∵ Standard equation of line is give as

y = m x + c

Where m is the slope of line and c is the y-intercept

Now, comparing given line equation with standard eq

So, The slope of the given line = m =
`(-5)/(2)`

Again,

The other line if passing through the points (- 2 , 3 ) And is parallel to given line

So, for parallel lines condition , the slope of both lines are equal

Let The slope of other line = M

So, M = m =
`(-5)/(2)`

∴ The equation of line with slope M and passing through points ( -2 , 3) is

y = M x + c

Now , satisfying the points

So, 3 =
`(-5)/(2)` × ( - 2 ) + c

or, 3 =
`(10)/(2)` + c

Or, 3 = 5 + c

∴ c = 3 - 5 = - 2

c = - 2

So, The equation of line with slope
`(-5)/(2)` and passing through points ( -2 , 3)

y =
`(-5)/(2)` x - 2

or, 2 y = - 5 x - 4

I.e 5 x + 2 y + 4 = 0

Similarly

The other line if passing through the points (- 2 , 3 ) And is perpendicular to given line

So, for perpendicular lines condition,the products of slope of both lines = - 1

Let The slope of other line = M'

So, M' × m = - 1

Or, M' ×
`(-5)/(2)` = - 1

Or, M' =
`(-1)/((-5)/(2))`

Or, M' =
`(2)/(5)`

∴ The equation of line with slope M and passing through points ( -2 , 3) is

y = M' x + c'

Now , satisfying the points

So, 3 =
`(2)/(5)` × ( - 2 ) + c'

or, 3 =
`(- 4)/(5)` + c'

Or, 3 × 5 = - 4 + 5× c'

∴ 5 c' = 15 + 4

or, 5 c' = 19

Or, c' =
`(19)/(5)`

So, The equation of line with slope
`(2)/(5)` and passing through points ( -2 , 3)

y =
`(2)/(5)` x +
`(19)/(5)`

y =
`(2 x + 19)/(5)`

Or, 5 y = 2 x + 19

Or, 2 x - 5 y + 19 = 0

Hence The equation of line parallel to given line and passing through points ( - 2 , 3 ) is 5 x + 2 y + 4 = 0

And The equation of line perpendicular to given line and passing through points ( - 2 , 3 ) is 2 x - 5 y + 19 = 0

Answer