Question

A line passes through `(8 ,2 )` and `(6 ,7 )`. A second line passes through `(3 ,8 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

Infinite points given by `(a,(31-5a)/2)`

Explanation:

As the first line passes through `(8,2)` and `(6,7)` its slope is

`(7-2)/(6-8)=-5/2`

Let the other point be `(a,b)` and as line joining `(a,b)` and `(3,8)` is parallel to first line, it will too have a slope of `-5/2`

Hence `(b-8)/(a-3)=-5/2`

or `2b-16=-5a+15`

i.e. `2b=-5a+31`

or `b=(31-5a)/2`

Hence the point `(a,(31-5a)/2)` is the other point.

Observe that by changing various values of `a`, we can have many points and hence therecould be infinite points. For example if `a=-3`, the point is `(-3,23)`.

graph{((x-8)^2+(y-2)^2-0.02)((x-6)^2+(y-7)^2-0.02)((x-3)^2+(y-8)^2-0.02)(5x+2y-31)(5x+2y-44)=0 [-6.38, 13.62, 0, 10]}