Question

8. Line ???? contains points (p − 4, 2) and (−2, 9). Line ???? contains points (p, −1) and (−1, 1).

a. Find the value of p if the lines are parallel.
b. Find the value(s) of p if the lines are perpendicular.

Answer 1

Answer with Step-by-step explanation:

We are given that a line passing through the point (p-4,2) and (-2,9) and other line passing through the point (p,-1) and (-1,1).

Slope-formula:
`m=(y_2-y_1)/(x_2-x_1)`

By using the formula

Slope of line which passing through the point (p-4,2) and (-2,9)

`m_1=(9-2)/(-2-p+4)=(7)/(2-p)`

Slope of other line which passing through the point (p,-1) and (-1,1)

`m_2=(1+1)/(-1-p)=(2)/(-1-p)`

When two lines are parallel then their slopes are equal

a.
`m_1=m_2`

`(7)/(2-p)=(2)/(-1-p)`

`-7-7p=4-2p`

`-7-4=-2p+7p`

`5p=-11`

`p=(-11)/(5)`

b.If the two lines are perpendicular then their slopes is opposite reciprocal to each other.

`m_1=-(1)/(m_2)`

`(7)/(2-p)=-(-1-p)/(2)`

`(7)/(2-p)=(1+p)/(2)`

`14=(1+p)(2-p)`

`14=2-p+2p-p^2`

`p^2+14=p+2`

`p^2-p+14-2\implies p^2-p+12=0`

It is quadratic equation in variable p

`D=b^2-4ac`

`D=(-1)^2-4(1)(12)=1-48=-47<0`

The root of equation are imaginary which is not possible .

If the lines are perpendicular then the value of p does not exist.