Question

A line segment is bisected by a line with the equation `2 y -4 x = 1`. If one end of the line segment is at `(3 ,8 )`, where is the other end?

Answer 1

The other end of the line segment is (4.2, 7.4)

Explanation:

Let the other end of the line segment be `(x_1,y_1)`
So the mid-point of `(3,8)` and `(x_1,y_1)` is `((x_1+3)/2,(y_1+8)/2)`
The mid point belongs to the line `2y-4x=1`
The slope of this line is`=2`

Sustituting the values of the mid point in this equation
`(2(y_1+8))/2-4((x_1+3))/2=1`
Simplifying the equation `y_1+8-2(x_1+3)=1`
`y_1+8-2x_1-6=1`
`y_1-2x_1=1-8+6=-1`

We need the equation of the line segment
the slope `m=-1/2` since the two lines are perpendicular and the product of the slopes is `m_1*m_2=-1`

so the equation is `(y_1-8)/(x_1-3)=-1/2`
`2y_1+x_1=19`
`y_1-2x_1=-1`
Solving we get `(y_1=2x_1-1)`
Substituting in other equation
`2(2x_1-1)+x_1=19`
`4x_1+x_1=21` `=>` `x_1=21/5=4.2`
and `y_1=-1+2*4.2=7.4`

So the point is (4.2,7.4)