Question

A line segment is bisected by a line with the equation `2 y + 9 x = 3`. If one end of the line segment is at `(3 ,2 )`, where is the other end?

Answer 1

The point `(-249/85,58/85)`

Explanation:

For a line with a slope that is perpendicular to the given line

`9x + 2y = 3" [1]"`

Swap the coefficients of x and y, change the sign of one of the coefficients, and set it equal to an arbitrary constant:

`2x - 9y = C`

To find the value of the constant, substitute the point `(3,2)` into the equation:

`2(3) - 9(2) = C`

`C = -12`

The equation of the bisected line is:

`2x - 9y = -12" [2]"`

Multiply equation [1] by 9 and equation [2] by 2:

`81x + 18y = 27" [3]"`
`4x - 18y = -24" [4]"`

Add equation [3] to equation [4]:

`85x = 3`

`x = 3/85`

This is the x coordinate of intersection.

Let `Deltax =` the change from the original x coordinate, 3, to the x coordinate of intersection:

`Deltax = (3/85 - 3) = -252/85`

Let `x_1 =` the x coordinate of the other end of the line segment.

`x_1 = 2Deltax + 3`

`x_1 = -504/85 + 3`

`x_1 = -249/85`

To find the corresponding y coordinate, substitute the value of `x_1` into equation [2]:

`2(-249/85) - 9y_1 = -12`

`y_1 = 58/85`