Question

The equation of three lines are given below

Line 1: 2y= 5x +6
Line 2: y=2/5x-1
Line 3: 10x+4y=-6
For each pair of lines, determine whether they are parallel perpendicular or neither

Answer 1

The line 2 and Line 3 are perpendicular to each others

But There is no relation between Line 1 and Line 2 , Line 1 and Line 3

Explanation:

Given as :

The three line equation is

Line 1 : 2 y = 5 x + 6

Line 2 : y =
`(2)/(5)` x - 1

Line 3 : 10 x + 4 y = - 6

Now, The standard equation of line is

y = m x + c

where m is the slope of line

And c is the y intercept

So, From Line 1

2 y = 5 x + 6

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

I.e y =
`(5)/(2)` x + 3

So, slope of this line =
`m_1` =
`(5)/(2)`

Again , From Line 2

y =
`(2)/(5)` x - 1

So, slope of this line =
`m_2` =
`(2)/(5)`

Similarly , From Line 3

10 x + 4 y = - 6

I.e 4 y = - 6 - 10 x

or, y = -
`(6)/(4)` -
`(10)/(4)` x

I.e y = -
`(5)/(2)` x -
`(6)/(4)`

So, Slope of this line =
`m_3` = -
`(5)/(2)`

Now, If the lines are parallel , then the slope of the lines are equal

And If the lines are perpendicular , then the product of the slopes of the lines = - 1

Now, From given lines

`m_2` ×
`m_3` =
`(2)/(5)` × ( -
`(5)/(2)` )

I.e
`m_2` ×
`m_3` = - 1

So, The line 2 and Line 3 are perpendicular to each others

But There is no relation between Line 1 and Line 2 , Line 1 and Line 3

Hence The line 2 and Line 3 are perpendicular to each others

But There is no relation between Line 1 and Line 2 , Line 1 and Line 3 Answer